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110
Molecular Simulation of Liquid Crystals Phase Equilibrium and the Solubility of Gases in Ordered Fluids

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Molecular Simulation of Liquid Crystals

Phase Equilibrium and the Solubilityof Gases in Ordered Fluids

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Molecular Simulation of Liquid Crystals

Phase Equilibrium and the Solubilityof Gases in Ordered Fluids

Proefschrift

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben,voorzitter van het College voor Promoties,

in het openbaar te verdedigen opvrijdag, 5 februari, 2016 om 12:30 uur

door

Bernardo Andrés OYARZÚN RIVERA

Diplom-Ingenieur, Technische Universtät Berlin,geboren te Viña del Mar, Chili.

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This dissertation has been approved by the

promotor: Prof. dr. ir. T.J.H. Vlugt

Composition of the doctoral committee:

Rector Magnificus, chairmanProf. dr. ir. T.J.H. Vlugt, Technische Universiteit Delft, promotor

Independent members:Dr. S.K. Schnell NTNU: Norges teknisk-naturvitenskapelige universitetProf. dr. C. Filippi Universiteit TwenteDr. D. Dubbeldam Universiteit van AmsterdamProf. dr. J. Meuldijk Technische Universiteit EindhovenProf. dr. ir. B.J. Boersma Technische Universiteit DelftProf. dr. D.J.E.M. Roekaerts Technische Universiteit Delft

This research was supported by the Stichting voor Technische Wetenschappen (Dutch Tech-nology Foundation, STW), applied science division of the Nederlandse organisatie voorWetenschappelijk Onderzoek (Netherlands Organization for Scientific Research, NWO)and the Technology Program of the Ministry of Economic Affairs. In addition, this workwas sponsored by the Stichting Nationale Computerfaciliteiten (National Computing Facil-ities Foundation, NCF) for the use of supercomputing facilities, with financial support fromNWO-EW.

Keywords: Liquid crystals, Molecular Simulation, Monte Carlo, Nematic, Gassolubility

Printed by: Wöhrmann Print Service

Front & Back: B.A. Oyarzún Rivera & I.N. Spînu

Copyright c© 2015 by B.A. Oyarzún Rivera

ISBN/EAN 978-94-6186-563-2

An electronic version of this dissertation is available athttp://repository.tudelft.nl/.

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This thesis is dedicated to my parentsfor all their love and support.

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Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Structured solvents for CO2 capture. . . . . . . . . . . . . . . . . . . . . 31.3 Molecular models for ordered phases . . . . . . . . . . . . . . . . . . . . 51.4 Molecular simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Scope and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Molecular simulation methods for chain fluids 112.1 Chain fluids and ordered phases . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Hard-sphere chain fluids . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Lennard-Jones chain fluids . . . . . . . . . . . . . . . . . . . . . 13

2.2 Order parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.1 Nematic order parameter . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Smectic order parameter . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Configurational-bias Monte Carlo . . . . . . . . . . . . . . . . . 142.3.2 NPT ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.3 Gibbs ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Expanded Gibbs ensemble . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 NVT expanded Gibbs ensemble . . . . . . . . . . . . . . . . . . . 232.4.2 NPT expanded Gibbs ensemble . . . . . . . . . . . . . . . . . . . 282.4.3 Wang-Landau algorithm . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Solubility of gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5.1 Henry’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5.2 Widom’s test-particle insertion method . . . . . . . . . . . . . . . 30

3 Liquid crystal phase behavior of hard-sphere chain fluids 333.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Linear hard-sphere chain fluids . . . . . . . . . . . . . . . . . . . . . . . 363.4 Partially flexible hard-sphere chain fluids . . . . . . . . . . . . . . . . . . 433.5 Solubility of hard spheres in hard-sphere chain fluids . . . . . . . . . . . . 473.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Isotropic-nematic phase equilibrium of hard-sphere chain fluids 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Linear hard-sphere chain molecules. . . . . . . . . . . . . . . . . . . . . 554.4 Binary mixtures of linear hard-sphere chain fluids. . . . . . . . . . . . . . 564.5 Binary mixtures of linear and partially-flexible hard-sphere chain fluids . . . 624.6 Solubility of hard spheres in hard-sphere chain fluids . . . . . . . . . . . . 644.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

vii

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viii Contents

5 Isotropic-nematic phase equilibrium of Lennard-Jones chain fluids 675.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 Simulation details and theory . . . . . . . . . . . . . . . . . . . . . . . . 695.3 Linear Lennard-Jones chains . . . . . . . . . . . . . . . . . . . . . . . . 705.4 Partially-flexible Lennard-Jones chains . . . . . . . . . . . . . . . . . . . 765.5 Binary mixture of linear Lennard-Jones chains . . . . . . . . . . . . . . . 765.6 Solubility of gases in Lennard-Jones chains . . . . . . . . . . . . . . . . . 795.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Bibliography 83

Summary 93

Samenvatting 95

Curriculum Vitæ 97

List of Journal Publications 99

Acknowledgement 101

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1Introduction

The phase behavior of liquid crystals and the solubility of gases in them are studied inthis thesis using molecular simulation techniques. The aim of this study is twofold: (1) toprovide a basic understanding of molecular principles behind the phase behavior of liquidcrystals and the solubility of gases in them, and (2) to contribute with state-of-the-art sim-ulation data on the liquid crystal phase behavior of chain fluids. The importance of thisstudy is rooted on the potential use of liquid crystals as solvents for CO2 capture. Resultsobtained in this work are used to understand the effect of molecular structure on gas sol-ubility. Furthermore, simulation data is used to validate a recently developed equation ofstate for nematic liquid crystal phases. The nature of liquid crystal phases and the prin-ciple behind their use as solvents for CO2 capture is shown in section 1.1 and section 1.2respectively. Molecular models used for the simulation of liquid crystals together with anintroduction to molecular simulation techniques are presented in section 1.4. Finally, theoutlook and scope of this thesis are described in section 1.5.

1

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2 1 Introduction

a) b) c) d)

Figure 1.1: Schematic representation of liquid crystal phases: a) Isotropic, b) Nematic, c) Smectic-A,d) Smectic-C.

1.1 BackgroundLiquid crystals are fluids with a certain degree of molecular order between the liquid andthe crystal state [1]. Liquid crystal phases are identified depending on the orientational andpositional order of their constituents. Fig. 1.1 shows schematically liquid crystal phaseswith their molecular ordering and classification. In the isotropic phase (Iso), molecules aredistributed randomly in space without any preferred long-range orientational nor positionalorder. In the nematic phase (Nem), molecules are oriented around a preferred direction butwith positions randomly distributed in space. The preferred direction that molecules followin the nematic phase is defined as the nematic director (n). In smectic phases, moleculesare distributed in layers showing both long-range orientational (nematic ordering) and posi-tional order. In smectic-A phases (SmA), the nematic director and the normal vector of thesmectic layers lay in parallel directions. In smectic-C phases (SmC) the nematic director istilted with respect to the normal vector of the smectic layers.

Molecular order in liquid crystal phases is a consequence of the balance between trans-lational and orientational entropy. In the transition from the isotropic to the nematic phase,the entropy loss by the formation of an orientational ordered phase is more than compen-sated by the gain in translational entropy. The orientational ordering of molecules in thenematic phase allows an increase in the number of accessible translational configurationsfor the molecules in the fluid. A similar argument can be used to explain the formation ofsmectic phases.

A necessary condition for the emergence of liquid crystal phases is shape anisotropy,i.e. molecular elongation and rigidity [1, 2]. Typically, liquid crystal molecules showingnematic and smectic phases have a backbone formed by two or more aromatic (or aliphatic)rings (rigid core) decorated with terminal groups which can vary from monoatomic sub-stituents to long alkyl chains (flexible tail) [2]. As an example, Fig. 1.2 shows the molecularstructure of the nematic forming 4-trans-4-pentyl-cyclohexyl-benzonitrile (PCH5).

The molecular order of liquid crystal phases confers specific physical properties to thesefluids. For instance, circularly birefringence of twisted nematic liquid crystals is widelyused for modulating light polarization in visual display devices [3]. Strong intermolecularinteractions in a nematic electron-donor liquid crystal fluid can improve charge mobilityin organic photovoltaic cells [4]. Ordering in liquid crystal complexes can modulate theimmune response in dendritic cells [5]. Of particular interest for this thesis is the solubilityof gases in liquid crystals and their use as solvents for gas separation applications [6–10].

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1.2 Structured solvents for CO2 capture

1

3

H3C

N

Figure 1.2: Liquid Crystal 4-(trans-4-Pentylcyclohexyl) Benzonitrile (PCH5). Transition temperatures at 1 atm:solid-nematic T S-N=303.05 K, nematic-isotropic T N-I=328.14 [11].

solvent cycle

Feed gas

Treated gas Separated gas

Figure 1.3: Schematic representation of an absorption-desorption process for the separation of gases withliquid crystal solvents. In the absorption stage (left), the gas of interest is separated from a mixture of gasesby the liquid crystal solvent. In the desorption stage (right), the liquid crystal solvent is cooled down for aphase transition to an ordered phase, releasing the separated gas. The depleted solvent is recirculated tothe absorption stage to complete the solvent cycle.

1.2 Structured solvents for CO2 captureThe fluid-fluid transition from the isotropic to the nematic phase causes a sudden drop in thesolubility of gases in liquid crystal fluids [7, 12, 13]. This phase change is associated with avery low enthalpy of transition ∆HN-I∼1−10 kJ/mol, taking place at a broad range of tem-perature and pressure conditions [14]. Moreover, liquid crystals are scarcely volatile withreported vapor pressures as low as 0.1−10 Pa at relatively high temperatures ∼373 K [15].Gas solubility and physical properties of liquid crystal fluids have attracted the attentionto their use as new solvents for CO2 capture processes [6, 7, 16]. Furthermore, the richphase behavior observed in liquid crystals fluids offers the potential of highly tailorablesolvents [14, 15].

Fig. 1.3 shows schematically the use of liquid crystals as solvents for gas separationin an absorption-desorption process. Absorption-desorption processes are widely used inthe chemical industry and represent the reference scheme for post-combustion CO2 captureprocesses [17, 18]. In the absorption stage, CO2 is separated from a gas mixture by theliquid crystal solvent in its isotropic state. In the desorption stage, a temperature drop causesa phase change of the liquid crystal fluid from its isotropic to the nematic state reducing thesolubility of the gas in the fluid. This temperature drop originates a two-phase region ofthe separated gas and the depleted liquid crystal solvent. The depleted solvent is heatedup and recirculated to the absorption column in its isotropic state completing the solvent

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4 1 Introduction

N+I

I

N

I+G

G

N+G

1 2

34

T

xk

Tabs

Tdes4'

TP

xdes xabs

Figure 1.4: Schematic phase diagram, temperature T vs. solute composition xk, for a gas solute k dilutedin a liquid crystal solvent. Isotropic (I), nematic (N) and gas phase (G) are single-phase regions. Two-phaseregions are identified with a plus sign. Numbers and dashed lines represent the solvent cycle of a simplifiedabsorption-desorption process: (1)-(2) absorption, (3)-(4) desorption, the gas is separated at (4’). Tabs is thetemperature in the absorption stage and Tdes is the temperature in the desorption stage. xabs is the molefraction of the solute in the loaded liquid crystal solvent after absorption. xdes is the mole fraction of the solutein the depleted liquid crystal solvent after desorption. TP is the gas-isotropic-nematic triple point.

cycle. The energy consumed in this process is principally due to: (1) cooling of the solventin the desorption stage for the phase transition from the isotropic to the nematic phaseQcooling; (2) heating up the solvent after the desorption stage for recovering the isotropicphase Qheating; and (3) the energy required for a pump to recirculate the solvent Wpump.The solvent performance is defined as the energy consumed in the process per amountof solvent used in the cycle (Qcooling + Qheating + Wpump)/msolvent [19]. It can be observedthat the total energy consumption, and therefore the solvent performance, depends directlyon the amount of solvent recirculated. A larger solubility difference of the gas betweenthe absorption (isotropic) and desorption (nematic) stages reduces the amount of solventemployed in the cycle, improving the performance of the solvent. The solubility differencebetween the isotropic and the nematic phase can be used as an indication of the performanceof liquid crystal solvents in an absoprtion-desorption process.

In Fig. 1.4, the absorption-desorption process is shown together with a schematic phasediagram for the mixture of gas solute with a liquid-crystal solvent. The solvent cycle isrepresented by the dashed lines. The stages of the solvent cycle are represented as follows:(1)-(2): the gas is absorbed in the isotropic liquid crystal solvent at constant temperatureTabs until the solvent is saturated with the gas at a concentration xabs. (2)-(3): the solventloaded with the gas is cooled down to the desorption temperature Tdes below the triplepoint (TP) to promote a phase split between a nematic and a gas phase. (3)-(4): the gas isseparated in the gas phase (4’) from the depleted liquid crystal solvent in the nematic phase(4) with a remaining gas concentration of xdes. (4)-(1): the depleted solvent is heated upfrom Tdes to Tabs for a phase change in the solvent from the nematic to the isotropic phaseand in that state it is recirculated back to the absorption stage. The solubility difference

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1.3 Molecular models for ordered phases

1

5

xabs - xdes is directly related to the performance of the solvent as stated before.Determining the solubility difference xabs−xdes would require the knowledge of a large

part of the phase diagram for a specific liquid crystal solvent and solute (as schematicallyrepresented in Fig. 1.4). However, a similar indication of the solvent performance can beobtained from the width of the isotropic-nematic region. In this work, the width of theisotropic-nematic region, i.e. the isotropic-nematic solubility difference, is used as a simplecriteria for comparing the potential solvent performance of different liquid crystal fluids.

1.3 Molecular models for ordered phasesA certain degree of shape anisotropy is a necessary molecular condition for the appearanceof ordered phases [1, 2]. Shape anisotropy introduces anisotropic repulsive and attractiveinteractions between molecules in a liquid crystal fluid. Repulsive interactions are responsi-ble for excluded volume effects while attractive interactions induce molecular aggregation.

Nematic ordering in a fluid was first predicted by Onsager as early as 1949 consideringanisotropic hard repulsive interactions only [20]. Almost a decade later, Maier and Saupedescribed the isotropic-nematic phase transition using an average angle-dependent attrac-tive potential derived from a mean-field approach [21–23]. In this mean-field approach re-pulsive forces are neglected assuming spherical symmetry in the distribution of the centersof mass. Quantitative deviations between theory and experiments have led to criticism onthe validity of the basic assumptions made in the Maier-Saupe theory [24–28]. Specifically,it has been shown that including angle dependent short-range repulsive interactions improvethe description of this mean-field approach [25, 29]. Therefore, it is assumed that a certaindegree of anisotropic repulsive interactions is essential for the appearance of liquid crystalphases [29–32]. Molecular simulation studies have shown the existence of liquid crystalphases in fluids made of molecules with a broad range of hard anisotropic shapes: harddisks [33, 34], hard ellipsoids [35–38], hard spherocylinders [39–44], cut hard spheres [45]and hard-sphere chains [46]. Frenkel [47] and Allen [48] provide excellent reviews onmolecular simulation of anisotropic hard molecules forming athermal liquid crystal phases.

Molecular models including both anisotropic attractive and repulsive interactions havebeen used in the simulation of thermotropic liquid crystals [49, 50]. From these, the Gay-Berne potential is a popular model based on an anisotropic form of the Lennard-Jones po-tential [51–56]. Other more elaborated anisotropic interaction models including attractiveand repulsive interactions are: hard-spherocylinder with an attractive square-well poten-tial [57–60], hard-spherocylinder with an attractive Lennard-Jones potential [61], hard-discwith an anisotropic square-well attractive potential [62], anisotropic soft-core spherocylin-der potential [63], and copolymers [64–66].

In this work, tangent chain molecules are used for the simulation of liquid crystal phasesas described in section 2.1. Chain molecules are molecules formed by a sequence of tangentbonded segments that interact with a defined form of the pair potential, e.g hard-sphere,Lennard-Jones or square-well potentials. The use of a segment-based approach is justifiedfrom a multi-scale description of fluids, connecting molecular information summarized ina coarse-grained model with a molecular-based theory for the description of liquid crystalphases. In a coarse-grained model, the information of the molecular structure (based onchemical groups, or small molecules, or in general a collection of atoms) is reduced to thepair potential parameters of a force-field [67, 68]. Simulations based on coarse-grained

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6 1 Introduction

models reach longer time and length scales than their atomistic counterparts, allowing abulk description of fluids. Moreover, molecular-based fluid theories as the family of theoriesderived from the statistical association fluid theory (SAFT) are based on a segment-baseddescription of molecules [69–76]. A common form of the interaction potential betweenboth theory and molecular simulations allow a direct comparison of theoretical results withsimulation data. Recently, an analytical equation of state describing nematic liquid crystalphases based on a SAFT approach was developed [77–79]. In this context, the study ofsegment-based chain molecules is meaningful in the sense that this type of model offersthe structural form required for a coarse-grained description of liquid crystal moleculesproviding, at the same time, a basis for validating the predictions obtained from a molecular-based equation of state.

1.4 Molecular simulationMolecular simulations are used to calculate the physical properties (macrostate) of a molec-ular system from statistical mechanics principles. A system is defined by a number N ofinteracting molecules at some defined thermodynamic state. The statistical nature of amolecular system determines that an observed macrostate is an average of a series of in-stantaneous states or microstates in which the system can exist. In a classical approach,a microstate is defined by the instantaneous set of positions and momenta (rN , pN) of allmolecules in the system, representing coordinates in the phase space P. The collection ofall possible microstates is identified as an ensemble.

The physical properties of a system at thermal equilibrium can be determined from theensemble average [80],

〈A〉 =

∫P

drNdpN exp[H(rN ,pN)/kBT

]A(rN ,pN)∫

PdrNdpN exp

[H(rN ,pN)/kBT

] . (1.1)

in which 〈A〉 stands for the ensemble average of an observable macroscopic propertyA(rN ,pN). The Hamiltonian H corresponds to the total energy of the system consideringthe potential and kinetic contributions. T is the temperature and kB is the Boltzmann factor.In Eq. 1.1, the integration over momenta can be carried out analytically and independentlyfrom the integration over positions since the forces between particles are momentum inde-pendent. Therefore, equilibrium properties of a system made of N molecules at constantvolume V and temperature T can be determined by the configurational average,

〈A〉 =

∫V drN exp

[−U(rN)/kBT

]A(rN)∫

V drN exp[−U(rN)/kBT

] . (1.2)

The Boltzmann factor exp[−U(rN)/kBT

]is proportional to the probability of the system to

be in a specific configuration rN as a function of the configuration’s potential energy U(rN)and temperature T .

The denominator of Eq. 1.2 is proportional to the partition function of the a system atconstant number molecules, volume, and temperature, i.e. the NVT ensemble,

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1.4 Molecular simulation

1

7

QNVT =1

Λ3N N!

∫V

drN exp[−U(rN)/kBT

], (1.3)

where Λ is the thermal de Broglie wavelength. The partition function is related to themacroscopic or thermodynamic properties of a system. In the case of the NVT ensemble,the partition function is directly related to the Helmholtz energy A(N,V,T ) of the system,

A(N,V,T ) = −kBT ln QNVT . (1.4)

All thermodynamic properties of a system could be in principle determined from Eq. 1.4,however, the integration over all possible microstates required by Eq. 1.3 is in practice im-possible. Random sampling over a finite space would in principle approximate macroscopicproperties calculated from configurational averages as in Eq. 1.2. This method is known asMonte Carlo simulation and is the principle that is used in this work for determining ther-modynamic properties. However, direct random sampling with equal probability for allconfigurations has the problem that most of the configurations have a very large energyand consequently a Boltzmann factor close to 0, resulting in an undefined division 0/0 forthe configurational average in Eq. 1.2. Metropolis et al. [81] provided a solution to thisproblem, proposing an algorithm in which configurations are generated with a probabilityproportional to the Boltzmann factor, sampling therefore the Boltzmann distribution:

“...the method we employ is actually a modified Monte Carlo scheme, where,instead of choosing configurations randomly, then weighting them withexp (−E/kBT ), we choose configurations with a probability exp (−E/kBT ) andweight them evenly.” [81]

Using the Metropolis sampling scheme, configurational averages can be calculated by,

〈A〉 =

n∑i=1

A(rN

i

)n

, (1.5)

where n is the number of configurations visited during sampling.Different configurations are obtained by performing molecular trial moves, e.g molecu-

lar translation or rotation. Every trial move is an attempt of changing the configuration ofthe system from a configuration α to β.

In Monte Carlo molecular simulations, obeying detailed balance is a sufficient conditionfor sampling a correct distribution [80, 82]. The condition of detailed balance is given by,

Pα Pmoveαβ Pacc

αβ = Pβ Pmoveβα Pacc

βα , (1.6)

where,

Pα : probability of the system to be in configuration α. In the NVT ensem-ble, Eq. 1.3, this probability is proportional to exp

[−U(rN)/kBT

].

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8 1 Introduction

Pmoveαβ : probability for attempting a change in the configuration of the system

from α to β.Paccαβ : probability that a change in the configuration of the system from α

to β is accepted.

The probabilities of the backward move from configuration β to α are analogous. Usuallythe probability of attempting a forward and backward move is selected to be the same,simplifying Eq. 1.6 to,

Pα Paccαβ = Pβ Pacc

βα . (1.7)

The criterion Cαβ for the transition from a configuration α to β is given by,

Cαβ =Paccαβ

Paccβα

=Pα

Pβ. (1.8)

In the Metropolis scheme the probability of accepting a change in the configuration ofthe system from α to β is chosen as follows [81],

Paccαβ = min

(1, Cαβ

). (1.9)

For accepting a trial move, a random number R obtained from an uniform distribution isgenerated. The probability that this number is less than Pacc

αβ is equal to Paccαβ . Therefore, a

trail move is accepted when R < Paccαβ . This method assures that microstates are sampled

following the Boltzmann distribution.The criterion for accepting a trial change from configuration α to β in the NVT ensemble

is given by,

Cαβ =Paccαβ

Paccβα

=Pα

Pβ= exp

[−

U(β) − U(α)kBT

]= exp

[−

∆UkBT

]. (1.10)

where U(α) and U(β) are the potential energies of configuration α and β, and ∆U = U(β)−U(α) is the energy difference between both configurations.

It can be observed that a configuration change, in the NVT ensemble, associated witha negative difference in the potential energy (∆U < 0) will be always accepted. Changescarrying out a positive difference in energy (∆U > 0) will be accepted with a probabilityproportional to exp [−∆U/kBT ].

1.5 Scope and outlineThe scope of this thesis is to contribute with simulation data to the understanding of thebehavior of liquid crystal phases and the solubility of gases in them. Linear and partially-flexible chain molecules are used for simulating the behavior of liquid crystal phases. Theeffect of molecular elongation is studied by linear chain molecules of different lengths andthe effect of flexibility is studied by partially-flexible molecules made of a linear and afreely-jointed part. The pair interaction between segments is accounted by hard-sphere andLennard-Jones potentials. The use of these potentials allows to study the effect of purely

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1

9

repulsive (excluded volume effects) and soft-attractive interactions (aggregation) on the liq-uid crystal behavior of chain fluids. The molecular model and pair interaction potentialsare presented in chapter 2 together with the simulation techniques used in this work. Chap-ter 2 includes a detailed description of the expanded Gibbs ensemble method used for thecalculation of liquid-crystal fluid phase equilibria. The liquid crystal behavior of singlecomponent hard-sphere chain fluids is reported in chapter 3. In this chapter, the effect ofmolecular elongation and flexibility on the phase behavior (packing fraction and pressure)and its relation with solubility is determined. Chapter 4 shows the isotropic-nematic phaseequilibria of single component and binary mixtures of hard-sphere chain fluids. The effectsof mixing on the coexisting isotropic-nematic phases and on the solubility of gases in themare determined in this chapter. Temperature effects for Lennard-Jones systems are studiedin chapter 5. In this chapter, isotropic-nematic phase equilibria are determined for singlecomponent systems with different degrees of molecular anisotropy and for a binary mix-ture of linear Lennard-Jones chains. Isotropic-nematic phase equilibria simulation resultsare compared with predictions obtained from the recently developed analytical equation ofstate by van Westen et al.: for hard-sphere chain fluids Refs. 77, 78, 83, and Lennard-Joneschain fluids Refs. 79, 84. Assumptions made in the development of the equation of state aredirectly evaluated by comparing analytical results with simulation data. A large amount ofsimulation data on the liquid crystal behavior of chain fluids and on the solubility of gasesin liquid crystal phases is presented in this work.

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2Molecular simulation methods for chain fluids

Linear and partially-flexible chain molecules are used to study the phase behavior of liq-uid crystal fluids using Monte Carlo simulations. The properties defining these molecu-lar models are introduced in section 2.1 together with the pair potentials defining theirinteractions. Ordered phases are identified by nematic and smectic order parameters asdescribed in section 2.2. Monte Carlo algorithms for the calculation of single-phase andtwo-phase systems are presented in section 2.3. Phase equilibria calculations of large chainmolecules are difficult to perform using classical methods due to the very low acceptancerate for the molecular exchange between phases. In section 2.4, an expanded version ofthe Gibbs ensemble is presented which improves the efficiency of phase equilibria calcu-lations compared to classical methods. Solubilities of gases in liquid crystal phases arecalculated using the Widom test-particle insertion method and are presented here as Henrycoefficients. Section 2.5 presents the Widom test-particle insertion method for determiningHenry coefficients.

11

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12 2 Molecular simulation methods for chain fluids

Figure 2.1: Partially-flexible chain molecule m−mR−mer made of a linear part (gray area) and a freely-jointedflexible part (white area). This specific model molecule is denoted as a 7-5-mer.

2.1 Chain fluids and ordered phasesThe molecular models used in this work are rigid linear chain and partially-flexible chainmolecules. A chain molecule is defined as a molecule made of spherical segments con-nected tangentially by a rigid bond with fixed length equal to the segment diameter σ. Alinear chain is a molecule with all segments laying on the same molecular axis. Partially-flexible molecules are made of a linear part and a freely-jointed flexible part as shown inFig. 2.1. The linear part introduces rigidity while the flexible part incorporates flexibility tothe model. The freely-jointed part does not have any bond-bending or torsional potential,therefore it is free to move between any possible molecular configuration limited only bythe rigid bond length constraint and the pair potential interaction between segments. Thepartially-flexible model is proposed in similarity with real liquid crystal molecules, formedby a rigid core and a flexible tail. The linear chain is a special case of the partially-flexiblemodel where no segments are present in the freely-jointed part. A linear chain with a num-ber of m segments is identified as a linear m-mer. A partially-flexible chain molecule madeof m segments in total and mR segments in the rigid part is denoted as a partially-flexiblem-mR-mer. Two types of pair interactions between segments, denoted here as ui j, are usedin this study: the hard-sphere potential (section 2.1.1), and the Lennard-Jones potential(section 2.1.2).

2.1.1. Hard-sphere chain fluidsHard-sphere segments are defined by a rigid sphere with diameter σ. Hard repulsions arethe only interactions present in this model with an infinite repulsion energy at distances ri j

lower than the segment diameter,

ui j =

{∞ ; ri j < σ0 ; ri j ≥ σ

. (2.1)

Vega et al. [85] studied the phase behavior of linear hard-sphere chain fluids with alength of 3, 4, 5, 6 and 7 segments from constant pressure Monte Carlo simulations. Theseauthors observed that hard-sphere chain fluids with a length of 5 segments and higher areable to form liquid crystal phases, while shorter chains experience a direct transition fromthe isotropic to the crystal state [85–87]. A more detailed simulation study on the phasebehavior of the linear hard-sphere 7-mer was performed by Williamson and Jackson [88].The isotropic-nematic phase behavior of the linear hard-sphere 8-mer and 20-mer chainfluids was obtained by Yethiraj and Fynewever [89]. Whittle and Masters [90] studied thephase behavior of 6 and 8 segment linear fused hard-sphere molecules with length-to-widthratios ranging from 3.5 to 5.2, observing a nematic phase only for the longest molecule.

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A partially-flexible model was employed by McBride and Vega et al. [91–93] for fusedhard-sphere molecules composed of a linear and a freely-jointed part with a total lengthof 15 beads and with 8 to 15 segments in the linear part, observing isotropic, nematic,and smectic phases, in all but the most flexible molecule where a nematic phase was notdetected. Escobedo and de Pablo [94] calculated the isotropic-nematic phase transition of8 and 16 hard-sphere chain molecules with finite and infinite (linear chains) bond-bendingpotentials.

2.1.2. Lennard-Jones chain fluidsThe Lennard-Jones pair potential between two segments separated a distance ri j is definedby,

ui j = 4ε

( σri j

)12

ri j

)6 , (2.2)

where ε is the depth of the potential well, and σ is the segment diameter defined as the dis-tance at which the potential between two segments is zero. Intermolecular pair interactionstake place between segments of different molecules and intramolecular pair interactionsbetween segments of the same molecule separated by two or more bonds. In this work,the Lennard-Jones pair potential is truncated at a cutoff radius rc = 2.5σ and the usuallong-range tail corrections are applied [37, 80].

Simulation results for the liquid crystal behavior of Lennard-Jones chains could notbe found in any other previous study. Galindo et al. [95] studied the phase behavior oflinear Lennard-Jones chains of 3 and 5 segments, however, no liquid crystalline phaseswere observed due to the short length of the chains.

2.2 Order parameters2.2.1. Nematic order parameterOrientational order is determined by the nematic order parameter and it is used for mon-itoring the transition from the isotropic to the nematic phase. It has a value of 0 in theisotropic fluid and experiences a near-step change at the isotropic-nematic phase transitionapproaching 1 in the nematic fluid.

The nematic order parameter S 2 is defined by the second-order Legendre polynomial [34],

S 2 =1N

⟨ N∑i=1

P2(cos θi)⟩

=1N

⟨ N∑i=1

(32

cos2 θi −12

)⟩, (2.3)

where θi is the angle between the molecular axis and the nematic director n of each moleculein the system. For partially-flexible molecules, the molecular axis is defined as the eigen-vector corresponding to the lowest eigenvalue of the moment of inertia tensor [89].

In molecular simulations when a nematic phase is formed, molecules do not necessaryalign across the largest dimension of the simulation box and knowledge a priori of thedirection of the nematic director is in general not possible. In practice, the orientationof the liquid crystal phase with respect to the laboratory framework is identified with theeigenvectors of the so-called de Gennes Q-tensor [1], Q = 1

N∑N

i=1(qiqi −13 I), where q is

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14 2 Molecular simulation methods for chain fluids

a unit vector identifying the direction of the molecular axis with respect to the laboratoryframe. In the case that two of the eigenvalues of Q are equal, an uniaxial nematic phase isidentified and the nematic director is associated with the eigenvector corresponding to thedifferent eigenvalue, i.e. the largest eigenvalue of Q. In the case that all three eigenvaluesare the same, the system is found in an isotropic state. For an uniaxial nematic phase,Q = S (n ⊗ n − 1

3 I), where n ⊗ n is the tensor product of the nematic director n. Theeigenvalues of the Q-tensor are λ+ = 2

3 S and λ0 = λ− = − 13 S , where S =

∑Ni=1( 3

2 cos2 θ− 12 )

[96]. The order parameter S 2 is estimated as S 2 = 32 〈λ+〉. It has to be noticed that the

Q tensor is diagonalized for every configuration considered in the ensemble average. Anaverage phase direction can also be obtained by the average 〈Q〉, whereby the Q tensor isdiagonalized only at the end of the simulation run. However, this approach requires thatfluctuations in the direction of the nematic director with respect to the laboratory frameare negligible, which is in principle not the general case and therefore this approach is notimplemented here. In this work, the order parameter is obtained from the ensemble average〈λ+〉 calculated every 103 Monte Carlo cycles.

2.2.2. Smectic order parameterPositional ordering is detected by the smectic order parameter, which is defined by themagnitude of the first Fourier component of the normalized density wave along the nematicdirector [97, 98],

τ =1N

⟨∣∣∣∣∣∣∣∣N∑

j=1

eikzz j

∣∣∣∣∣∣∣∣⟩. (2.4)

where N is the total number of molecules and kz=2π/λz. The periodicity of the smectic lay-ers is λz and z j are the coordinates of the center of mass of the j-th molecule in the directionof the nematic director. Values of the smectic order parameter, which differ significantlyfrom zero indicate the presence of smectic layers.

2.3 Monte Carlo simulationsIn this section classical methods for the simulation of fluid systems are presented. The gen-eral principles of Monte Carlo simulation methods were presented in section 1.4 togetherwith the isobaric-isothermal NVT ensemble. Changes in the configuration of partially-flexible chain molecules are obtained using the configurational-bias method. The isobaric-isothermal NPT ensemble is described for the calculation of properties of single phasesystems. Direct phase equilibrium calculations of simple fluids are performed in the Gibbs-ensemble. The methods and algorithms presented here are general independent of the formof the pair interaction potential. The specific cases of hard-sphere or Lennard-Jones poten-tials are obtained by replacing the form of the interaction potential ui j by Eq. 2.1 or Eq. 2.2respectively.

2.3.1. Configurational-bias Monte CarloIn the configurational-bias method [99–102], the configuration of a partially-flexible moleculeis changed by regrowing the freely-jointed part attempting a change from configuration α to

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β. In this method, the flexible part is regrowth segment by segment attempting each time anumber of trail orientations with coordinates {b1, ...,bk} for every regrown segment. For thechain molecules with fixed bond length studied here, the trial orientations are distributedaround a sphere with radius equal to the segment diameter. Each trail orientation experi-ences pair interactions with the other segments of the same molecule (excepting the segmentfrom which the trail segment is regrown from) and with the segments of all other moleculesin the system. For the actual regrown segment i, the Boltzmann factor is calculated for eachtrail orientation k, and one of those, denoted by n, is selected with a probability,

pi(bn) =exp [−ui(bn)/kBT ]

k∑j=1

exp[−ui(b j)/kBT

] . (2.5)

For each regrown segment, the statistical weight of the segment in the regrowing processof the flexible part is the denominator of Eq. 2.5,

wi =

k∑j=1

exp[−ui(b j)/kBT

]. (2.6)

In partially-flexible molecules, only the configuration of the flexible part is changed, i.efrom segment mR + 1 to m. After the whole chain is regrown, the Rosenbluth factor of thenew configuration β is obtained,

Wβ =

m∏i=mR+1

wi. (2.7)

The Rosenbluth factor of the old configuration α is also calculated using the old con-figuration as the first trial direction. Considering the condition of microscopic reversibility,Eq. 1.7, the criterion for accepting a change in the configuration of a chain molecule byconfigurational-bias regrowth is given by,

Cαβ =Wα

Wβ. (2.8)

A trial change in the configuration of a chain molecule from α to β is accepted with aprobability given by Eq. 1.9.

2.3.2. NPT ensembleThe phase behavior of single phase system made of N molecules at constant pressure P andtemperature T are obtained in the isobaric-isothermal NPT ensemble [80],

QNPT =

(P

kBT

)1

Λ3N N!

∫ ∞

0dV VN exp

[−

PVkBT

] ∫ 1

0dsN exp

[−

U(sN)kBT

]. (2.9)

The scaled coordinates, s = r/L where L = V1/3, are introduced here to perform arandom walk in a sampling space that is independent of the actual size of the simulation

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16 2 Molecular simulation methods for chain fluids

box. These coordinates refer to the position of every segment of all molecules in the systemaccounting therefore for the molecular configuration and orientation of molecules. Thepartition function of the NPT ensemble for configurational changes in the logarithm of thevolume V is derived as follows,

QNPT =

(P

kBT

)1

Λ3N N!

∫ ∞

−∞

d ln V VN+1 exp[−

PVkBT

] ∫ 1

0dsN exp

[−

U(sN)kBT

]. (2.10)

The probability that the system is in a determined configuration α with volume ln V andcoordinates sN is proportional to,

Pα ∝ VN+1 exp[−

PVkBT

]exp

[−

U(sN)kBT

]. (2.11)

Changes in the configuration of the system are performed by trial changes in the posi-tion/orientation of molecules, the molecular configuration of partially-flexible molecules,and the volume of the system (the size of the simulation box). Trial changes that do notinvolve a change in volume of the system are equivalent to a change in the NVT ensembleand the acceptance criterion for a trail change from a configuration α to a configuration β isequivalent to the one presented in section 1.4,

Cαβ = exp[−

∆UkBT

], (1.10)

where ∆U is the difference in potential energy between both configurations.

Volume changeThe acceptance criterion for a trial change in ln V is given by,

Cαβ = exp[

P∆VkBT

+ (N + 1) ln(

V + ∆VV

)−

∆UkBT

]. (2.12)

where ∆V is the change in the volume of the system.

2.3.3. Gibbs ensembleDirect phase equilibria simulations are performed in the method proposed by Panagiotopou-los [103, 104], denoted as the Gibbs ensemble. The basis of this method are presented inthis section as a preamble of the expanded Gibbs ensemble simulation method developedin the next section. In the Gibbs ensemble method two phases, subsystems a and b, arebrought into contact to achieve thermal, mechanical, and chemical potential equilibrium:

Ta = Tb = TPa = Pb (= P)µa,i = µb,i.

(2.13)

Two forms of the method can be identified: simulations at constant total volume V andtemperature T conditions (NVT Gibbs ensemble), or simulations at constant pressure P andtemperature T conditions (NPT Gibbs ensemble). In both cases, simulations are performed

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for a total number of molecules N = Na + Nb.The NVT Gibbs ensemble is used to calculate the phase equilibria of pure component

systems. In this case, only one thermodynamic variable, the temperature, is required fordetermining the thermodynamic state of the system. The total volume of the system V =

Va + Vb is selected in order to match a density in the two-phase region. The volume of oneof the phases is varied independently until both phases reach the same pressure.

The NPT version of the Gibbs ensemble is used in the case of mixtures where two ther-modynamic variables, in this case temperature and pressure, are required for determiningthe thermodynamic state of the system.

In both cases, thermal equilibration is achieved by rotation/displacement of moleculesand partial-regrowth of partially-flexible molecules. Mechanical equilibrium is obtainedby: volume changes of one subsystem in the NVT Gibbs ensemble, or by volume changesin both subsystems in the NPT Gibbs ensemble. Equality in the chemical potential of everycomponent i is achieved by molecular exchange between both subsystems.

NVT Gibbs ensembleThe partition function of the NVT Gibbs ensemble is given by [80],

QNVT−Gibbs =1

N!Λ3NV

N∑Na=0

N!Na!Nb!

∫ V

0dVaVNa

a VNbb

×

∫ 1

0dsNa

a exp[−

Ua(sNaa )

kBT

] ∫ 1

0dsNb

b exp

−Ub(sNbb )

kBT

. (2.14)

Similarly as in the previous section, reduced coordinates are introduced, s j = r j/L j

where L j = V1/3j is the box length of either subsystem a or b. Na and Nb are the number of

molecules in each subsystem.The probability of finding the system in a configuration α with coordinates (sNa

a , sNbb )

and volume ratio Va and Vb is proportional to,

Pα ∝N!

Na!Nb!VNa

a VNbb

Vexp

[−

Ua(sNaa )

kBT

]exp

−Ub(sNbb )

kBT

. (2.15)

Volume changeVolume changes are performed by a random walk in the logarithm of the ratio between thevolume of both subsystems ln(Va/Vb). The partition function changes then to,

QNVT−Gibbs =1

N!Λ3NV

N∑Na=0

N!Na!Nb!

∫ ∞

−∞

d ln(

Va

Vb

)VaVb

VVNa

a VNbb

×

∫ 1

0dsNa

a exp[−

Ua(sNaa )

kBT

] ∫ 1

0dsNb

b exp

−Ub(sNbb )

kBT

. (2.16)

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18 2 Molecular simulation methods for chain fluids

The acceptance criterion for a trial change in the configuration of the system from α toβ by a volume change trial move in the phase space ln(Va/Vb) = ln(Va/(V − Va)) is thengiven by,

Cαβ = exp[(Na + 1) ln

(Va + ∆Va

Va

)+ (Nb + 1) ln

(Vb − ∆Va

Vb

)−

∆Ua

kBT−

∆Ub

kBT

], (2.17)

where ∆Va indicates the magnitude of the volume change in subsystem a, and ∆Ua and ∆Ub

indicate the related energy change in both subsystems.

Algorithm for molecular exchange in the Gibbs ensembleThe algorithm for molecular exchange as originally proposed by Panagiotopoulos [103] isused in this work. The algorithm is described as follows:

1. The molecular donor and recipient subsystems are first chosen randomly,P sub a = P sub b = 1/2.

2. The component to be exchanged is chosen randomly with same probabilityfor all components in the mixture, Pcomp 1 = Pcomp 2 = ... = Pcomp i.

3. A random molecule of component i in the donor subsystem (a in this case) isselected with probability, Pmol i,a

α = 1/Na,i. If Na,i = 0, then the trail move isimmediately rejected.

3. The molecule is transferred to a random position in the recipient subsystem(b in this case).

4. A new configuration β is accepted with a probability given by min(1, Cαβ).

A different form of the algorithm satisfying detailed balance can be obtained by select-ing first the molecule to exchange regardless the subsystem in which it is in. This choiceleads to a different acceptance criterion for the molecular exchange move as the one pre-sented in this section [105].

Molecular exchange, pure componentsThe transfer of a particle between both subsystems will transform the overall configurationof the system from configuration α to configuration β. The probability of such a transfer isthen equal to,

Pαβ = Na!Nb! Pα P sub, a Pmol, aα Ppos, b Pacc

αβ , (2.18)

where,

Pαβ : probability to change the configuration of the system from α to β.Pα : probability of the system to be in configuration α, Eq. 2.15.P sub, a : probability of choosing subsystem a as molecular donor.Pmol, aα : probability that a specific molecule is chosen from the Na molecules

in subsystem a in configuration α.Ppos, b : probability of selecting a position for insertion in subsystem b.Paccαβ : acceptance criterion for the particle transfer.

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The probability of the backward move, i.e., the probability of a particle exchange fromsubsystem b to a and therefore to change the configuration of the system from α to β isequal to,

Pβα = (Na − 1)!(Nb + 1)! Pβ P sub, b Pmol, bβ Ppos, a Pacc

βα . (2.19)

The probability of the total system to be in configuration β is given by,

Pβ ∝N!

(Na − 1)!(Nb + 1)!VNa−1

a VNb+1b

V

× exp−Ua(s(Na−1)

a )kBT

exp

−Ub(s(Nb+1)b )

kBT

. (2.20)

The condition of microscopic reversibility states that the probability of the forward Pαβ

and backward Pβα move has to be the same,

Pαβ = Pβα. (2.21)

The probability of choosing any of both subsystem as molecular donor is the same, thereforeP sub, a = P sub, b = 1/2. The probability of choosing a molecule from subsystem a inconfiguration α is proportional to Pmol, a

α = 1/Na for the forward move and the probabilityof choosing a molecule in subsystem b is proportional to Pmol, b

β = 1/(Nb + 1) for thebackward move. The probability of choosing a position for the insertion of a molecule isequal for both subsystems Ppos, a = Ppos, b. Considering this, from Eq 2.18 and Eq. 2.19,the following acceptance criterion is derived from Eq. 2.21,

Cαβ =Paccαβ

Paccβα

=(Na − 1)!(Nb + 1)!

Na!Nb!P sub, b

P sub, a

Ppart, bβ

Ppart, aα

Ppos, a

Ppos, b

= exp[ln

(NaVb

(Nb + 1)Va

)−

∆Ua

kBT−

∆Ub

kBT

], (2.22)

where ∆Ua and ∆Ub are the change in energy in both subsystems due to molecular ex-change.

NPT Gibbs ensembleThe NPT version of the Gibbs ensemble is used for the calculation of the phase equilibriaof mixtures. Here, the partition function and algorithms are developed for the case of abinary mixture. The general case of a multicomponent mixture is presented at the end ofthe section. The partition function for phase equilibria calculations in the Gibbs ensembleat constant number of molecules N, pressure P and temperature T , for a binary mixture ofcomponents 1 and 2 is given by,

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20 2 Molecular simulation methods for chain fluids

QNPT−Gibbs =1

N1!Λ3N1 V0

1N2!Λ3N2 V0

N1∑Na,1=0

N1!Na,1!Nb,1!

N2∑Na,2=0

N2!Na,2!Nb,2!

×

∫ ∞

0dVaVNa

a exp[−

PVa

kBT

] ∫ ∞

0dVbVNb

b exp[−

PVb

kBT

]

×

∫ 1

0dsNa,1

a,1

∫ 1

0dsNa,2

a,2 exp

−Ua(sNa,1

a,1 , sNa,2

a,2 )

kBT

×

∫ 1

0dsNb,1

b,1

∫ 1

0dsNb,2

b,2 exp

−Ub(sNb,1

b,1 , sNb,2

b,2 )

kBT

, (2.23)

where N1 = Na,1 + Nb,1 is the total number of molecules of component 1, and Na,1 and Nb,1are the number of molecules of component 1 present in subsystem a and b respectively. Asbefore, reduced coordinates are introduced, where sNa,1

a,1 are the coordinates of the segmentsof all molecules of component 1 in subsystem a. All other coordinates are represented in asimilar manner. V0 is a reference volume introduced here for making the partition functiondimensionless.

The probability that the system is in configuration α is proportional to,

Pα ∝ expln N1!

Na1 !Nb

1 !

+ ln N2!

Na2 !Nb

2 !

+ Na ln Va + Nb ln Vb

−PVa

kBT−

PVb

kBT−

Ua(sNa,1

a,1 , sNa,2

a,2 )

kBT−

Ub(sNb,1

b,1 , sNb,2

b,2 )

kBT

. (2.24)

Volume changeVolume changes are performed in an independent manner in both subsystems. As before,a random walk in the logarithm of the volume is preferred and the criterion for accepting atrial change in ln V j, where j is either subsystem a or b, is equivalent to Eq. 2.12.

Molecular exchange, binary mixturesThe equality of chemical potentials is obtained by the exchange of particles of component 1and 2 between both subsystems a and b at constant temperature and pressure. The transferof a molecule, e.g. of component 1, between both subsystems will transform the overallconfiguration of the system from configuration α to configuration β. The probability ofsuch a transfer is then proportional to,

Pαβ = Na,1!Nb,1!Na,2!Nb,2! P sub a Pcomp 1 Pmol 1,aα Ppos b Pα Pacc

αβ , (2.25)

where,

Pαβ : probability of changing the overall configuration of the system fromα to β due to molecular exchange.

P sub a : probability of choosing subsystem a as particle donor.Pcomp 1 : probability of choosing component 1 for molecular exchange.

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21

Pmol 1,aα : probability that a specific molecule of component 1 is chosen from

the Na,1 molecules in subsystem a.Ppos b : probability of selecting a position for insertion in subsystem b.Pα : probability of the total system to be in configuration α, Eq 2.24.Paccαβ : acceptance criterion for the particle transfer.

The probability of the reverse move, i.e. the probability of transferring a moleculeof component 1 from subsystem b to a, changing therefore the configuration of the systemfrom configuration α to β, is equal to,

Pβα = (Na,1 − 1)!(Nb,1 + 1)!Na,2!Nb,2!

× P sub b Pcomp 1 Pmol 1,bβ Ppos a Pβ Pacc

βα , (2.26)

where now,

Pβ ∝ exp[ln

(N1!

(Na,1 − 1)!(Nb,1 + 1)!

)+ ln

(N2!

Na,2!Nb,2!

)+ Na ln Va + (Nb + 2) ln Vb −

PVa

kBT

−PVb

kBT−

Ua(sNa,1−1a,1 , sNa,2

a,2 )

kBT−

Ub(sNb,1+1b,1 , sNb,2

b,2 )

kBT

. (2.27)

The acceptance criterion Cαβ is obtained from the condition of microscopic reversibilityEq. 2.21. The probability of choosing any of both subsystems is the same P sub a = P sub b =

1/2. The probability of choosing component 1 or 2 for molecular exchange is also sym-metric Pcomp 1 = Pcomp 1 = 1/2. The probability of choosing a molecule of component1 in subsystem a in configuration α is Pmol a

α = 1/Na,1 and the probability of choosing amolecule of component 1 in subsystem b in configuration β is Pmol b

α = 1/(Nb,1 + 1). IfNa,1 = 0 then the trial move is immediately rejected. The probability of selecting a posi-tion for insertion is the same in both subsystems Ppos a = Ppos b. Therefore, the acceptancecriterion for a trial move involving molecular exchange is given by,

Cαβ =Paccαβ

Paccβα

=(Na,1 − 1)!(Nb,1 − 1)!Na,2!Nb,2!

Na,1!Nb,1!Na,2!Nb,2!P sub b

P sub a

Pcomp 1

Pcomp 1

Pmol 1,bβ

Pmol 1,aα

Ppos a

Ppos b

= exp[ln

(Na,1Vb

(Nb,1 + 1)Va

)−

∆Ua

kBT−

∆Ub

kBT

]. (2.28)

∆Ua and ∆Ub are the changes in energy in both subsystems due to the molecular exchangeof a molecule of component i from subsystem a to b.

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22 2 Molecular simulation methods for chain fluids

Molecular exchange, multicomponent mixturesAfter inspection of Eq. 2.28 and Eq. 2.22, the following acceptance criterion can be deducedfor the molecular exchange of any component i in a multicomponent system following theselection algorithm described in the beginning of this section,

Cαβ = exp[ln

(Na,iVb

(Nb,i + 1)Va

)−

∆Ua

kBT−

∆Ub

kBT

]. (2.29)

2.4 Expanded Gibbs ensemblePhase equilibria of simple systems can be directly calculated using the Gibbs ensemble asshown in the previous section. In Gibbs ensemble simulations, the equilibrium of chemi-cal potentials is achieved by the exchange of whole molecules between both phases. Whiletransfer of whole molecules between phases is effective only for short and simple molecules,highly anisotropic or complex molecules have a very low probability of transfer accep-tance [101, 104, 106]. This condition results in poor ergodic sampling of systems made ofnon-simple molecules within reasonable simulation time. Advanced techniques have beendeveloped to overcome this difficulty. Configurational-bias sampling [100, 107] was im-plemented in the Gibbs ensemble for improving the simulation of vapor-liquid equilibriaof non-simple fluids [102, 108–111]. Although configurational-bias Monte Carlo showsan improved molecular transfer efficiency over the traditional Gibbs ensemble scheme,it is computational expensive and its efficiency is decreased as molecular complexity in-creases [112–114]. Even improvements of the configurational-bias Monte Carlo techniqueas the Recoil Growth method show to be inefficient for dense systems [115, 116].

Either in the traditional or in the configurational-bias implementation of the Gibbs en-semble, molecular transfers are attempted by insertion/deletion of whole molecules. In-tuitively, molecular transfer efficiency can be improved by attempting at every step thetransfer of molecular segments rather than of whole molecules. Expanded ensemble tech-niques are based on this principle [117–119]. In these methods, the ensemble of a systemof whole molecules is expanded into a series of sub-ensembles covering for one fractionalmolecule, the range between a “ghost” molecule (a molecule without any intermolecularinteractions) to a fully coupled molecule (a molecule where all intermolecular interactionsare present). In expanded Gibbs ensemble simulations, molecular transfer between phasesis achieved by gradually coupling a fractional molecule in one phase while, at the sametime, a complementary fractional molecule is decoupled from the other [120, 121]. Grad-ual coupling/decoupling is performed by a random walk over sub-ensembles, each one ofthem corresponding to a defined fractional state that determines the degree of intermolecularcoupling of the fractional molecules. In the traditional Metropolis sampling scheme [81], arandom walk over fractional states results in an uneven distribution of the relative probabil-ity of visiting fractional states. This condition restricts molecular transfer between phases,limiting the efficiency of the method. A smooth transition between all fractional states isdesired, striving ideally to the same relative probability for visiting any fractional state. Forthis aim, the partition function of the expanded ensemble is modified by a weight functionfor each fractional state, changing the Boltzmann statistics of the original system [122–124]. Numerical values for these weight functions are not known a priori and an iterativemethod for determining them is required, see section 2.4.3 and Refs. [117–119].

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2

23

Simulation of chain molecules are performed in this study in an expanded version of theGibbs ensemble [125]. The method is based on the gradual exchange of molecules betweenphases by the coordinated coupling/decoupling of segments of a fractional molecule . Thereis one fractional molecule present in each subsystem for each component. The simulationmethod is based on the ideas of Lyubartsev et. al. [117, 126, 127] on expanded ensemblesand is similar in spirit to the Continuous Fractional Component Monte Carlo method ofMaginn and co-workers [121, 128–140]. Instead of the continuous insertion presented intheir work, here a segment-wise insertion of chain molecules is proposed. Similarly, Es-cobedo and de Pablo developed the expanded ensemble ideas for the calculation of phaseequilibria of polymer molecules [119, 120, 141]. However, in their method, configurational-bias is used for stepwise insertion/deletion of molecular segments. Furthermore, in thatmethod, fractional molecules in their end-states are counted as whole molecules, changingthe form of the partition function every time an end-state is visited [119]. In the methodpresented here, the total number of whole molecules and therefore the form of the partitionfunction remain unchanged during the simulation.

Constant volume expanded Gibbs ensemble simulations are used for determining thephase equilibria of pure components. A constant pressure formulation of the expandedGibbs ensemble is used for the calculation of the phase equilibria of binary mixtures.

2.4.1. NVT expanded Gibbs ensembleThe partition function for a multicomponent system made of n components with a totalnumber of N whole molecules and one fractional molecule per component in each subsys-tem, a and b, for a constant total volume V = Va + Vb, at constant temperature T , is givenby,

QNVT−EG =1

Λ3(N+2n)V

n∏i=1

1Ni!

Ni∑Na,i=0

mi∑λi=0

Ni!Na,i!Nb,i!

exp [wi(λi)]

×

∫ V

0dVa (Va)(Na+n) (Vb)(Nb+n)

×

∫ 1

0dsNa

a dsna exp

−Ua(sNaa , sn

a, λ1, ..., λn)kBT

×

∫ 1

0dsNb

b dsnb exp

−Ub(sNbb , sn

b, λ1, ..., λn)kBT

. (2.30)

Here Λ is the de Broglie wavelength and kB is the Boltzmann factor. Ni = Na,i + Nb,i is thetotal number of whole molecules of component i, and Na =

∑ni Na,i and Nb =

∑ni Nb,i are the

total number of whole molecules in subsystem a and b respectively. A molecule is definedby a chain of interacting segments with a total length mi for a molecule of component i. Thedimensionless coordinates sNa

a and sNbb describe the positions of all segments of the Na and

Nb whole molecules. The dimensionless coordinates sna and sn

b describe the positions of allbeads of the n fractional molecules in subsystem a and b respectively. Each subsystem hasone fractional molecule per component with fractional states defined by the coupling pa-rameter λi. The coupling parameter determines the fractional state (number of interacting

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24 2 Molecular simulation methods for chain fluids

segments) of the fractional molecules in both subsystems simultaneously, i.e. λi coupledsegments in subsystem a and (mi − λi) in subsystem b. Therefore, λi = 0 defines an idealchain molecule in subsystem a (a molecule where only bonded interactions are present) anda fractional molecule identical (but not equal) to a whole molecule in subsystem b. Here,discrete values of the coupling parameter in the range λi = [0, ...,mi] are considered, leadingto a total of mi + 1 possible fractional states. Note that only whole molecules of a specificcomponent are indistinguishable from each other and can be permuted between subsystems.This explains the factors 1/Ni! and Ni!/Na,i!Nb,i! expressed in terms of the number of wholemolecules. The weight functions wi(λi) are introduced to modify the Boltzmann statisticsof the system in order to improve the sampling efficiency of all fractional states. The weightfunctions of each component are considered to be independent of each other, which is exactin the thermodynamic limit. Weight functions are determined iteratively using the Wang-Landau sampling method [142, 143] as explained below. The total energy of a subsystemUa (the same for subsystem b), is the sum of the bonded interactions Ubond

a for all moleculesin the subsystem, plus the intermolecular U inter

a (sNaa , sn

a, λ1, ..., λn) and intramolecular interac-tions U intra

a (sNaa , sn

a, λ1, ..., λn) for the Na whole and n fractional molecules in the subsystem,Ua = Unon−bonded

a + Ubondeda = U inter

a + U intraa + Ubond

a . Only non-bonded interactions (inter-molecular and intramolecular) are a function of the fractional state. Bonded interactions donot depend on the fractional state and are equivalent to those of whole molecules.

The probability that the system is in configuration α is proportional to,

Pα ∝ exp

n∑i=1

(ln

(Ni!

Na,i!Nb,i!

)+ wi(λi)

)+(Na + n) ln Va + (Nb + n) ln Vb

−Ua(sNa

a , sna, λ1, ..., λn)

kBT−

Ub(sNbb , sn

b, λ1, ..., λn)kBT

. (2.31)

A Markow chain in the space of fractional states can be organized by random changesin the coupling parameter, λi. Two different cases can be identified: changes withoutmolecule transfer, and changes with molecule transfer. A coupling parameter change with-out molecule transfer will occur when the new fractional state λnew

i = λoldi + ∆λi has a value

within the range [0, ...,mi] and a change with molecule transfer takes place when λnewi is

outside this range. The end-states, λi = 0 and λi = mi, deserve special attention. A frac-tional molecule with coupling parameter λi = mi is fully coupled to the system, however,this molecule will become equal to a whole one only when a further change in the fractionalstate takes place (see Fig. 2.2). A molecule transfer is therefore defined as the state transi-tion λi = mi → 0 from the old to a new randomly inserted fractional molecule. Strictly, onlyconfigurations with fractional molecules in their end-states have a clear physical meaningequivalent to that of a system without fractional molecules. However, sampling only whenan end-state is visited has the inconvenience of observables averaged over a reduced numberof samples. For a pure component system this is not truly a limitation, but for multicompo-nent systems the probability of visiting an end-state for all components at the same time isreduced to the joint probability of visiting those states. Nevertheless, in the thermodynamiclimit, the fractional state does not affect the properties of the system. Therefore, all thermo-

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2.4 Expanded Gibbs ensemble

2

25

...

Ni Ni+1

mi10 2

...... mi10 2

...

λ - space

Figure 2.2: Schematic representation of changes in the coupling parameter space λ for component i. When achange in λi reaches the end-state mi, the fractional molecule has the same interactions as a whole moleculebut it is still not considered as a whole one. A new whole molecule is transferred to the subsystem, Ni + 1,only when a further change in the coupling parameter reaches a state beyond the full fractional state mi.

dynamic properties are calculated based on the number of whole molecules present in thesystem independent of the fractional state.

Volume changeAs in the case of the Gibbs ensemble, volume changes are performed by a random walkin the logarithm of the relative volume ln[Va/Vb]. Taking this into account, the partitionfunction takes the form,

QNVT−EG =1

Λ3(N+2n)V

n∏i=1

1Ni!

Ni∑Na,i=0

mi∑λi=0

Ni!Na,i!Nb,i!

exp [wi(λi)]

×

∫ ∞

−∞

d ln(

Va

Vb

)VaVb

V(Va)Na+n(Vb)Nb+n

×

∫ 1

0dsNa

a dsna exp

−Ua(sNaa , sn

a, λ1, ..., λn)kBT

×

∫ 1

0dsNb

b dsnb exp

−Ub(sNbb , sn

b, λ1, ..., λn)kBT

. (2.32)

The criterion for a reversible change in the volume by a random walk in ln[Va/(V−Va)]is then equal to,

Cαβ = exp[(Na + n + 1) ln

(Va + ∆Va

Va

)+ (Nb + n + 1) ln

(Vb − ∆Va

Vb

)

−∆Ua

kBT−

∆Ub

kBT

], (2.33)

where ∆Ua and ∆Ub are the change in energy related to the volume change in in bothsubsystems.

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26 2 Molecular simulation methods for chain fluids

Algorithm for a coupling parameter change in the expanded Gibbs ensembleThe algorithm for performing a trail change in the coupling parameter is described as fol-lows:

1. The subsystem in which the fractional state is incremented is first chosenrandomly with equal probability, P sub a = P sub b = 1/2.

2. The component i which experiences a change in the coupling parameter ischosen randomly with same probability for all components in the mixture,Pcomp 1 = Pcomp 2 = ... = Pcomp i.

3. A positive discrete change in the coupling parameter ∆λi is chosen randomlyfrom [1, 2, ...,∆λmax

i ], where ∆λmaxi is adjusted depending on the acceptance

ratio required for this trial move.4. If the coupling parameter change ∆λi is not related with a molecular exchange,

the criterion for accepting a trial change in the coupling parameter is given byEq. 2.36.

5. If the coupling parameter change ∆λi is related with a molecular exchange,a new fractional molecule with random configuration is inserted randomly inthe subsystem where the coupling parameter is increased and the the criterionfor accepting a trial change in the coupling parameter is given by Eq. 2.39.

6. A new configuration β is accepted with a probability given by min(1, Cαβ).

The acceptance criterion for a change in the coupling parameter without and withmolecular exchange is derived based on this form of the algorithm.

Coupling parameter change without molecular exchangeEvery attempt of changing the configuration of the system from a configuration α to aconfiguration β by a change in the coupling parameter ∆λi for component i has to satisfythe condition of microscopic reversibility [105],

Pαβ = Pβα. (2.34)

For a change in the coupling parameter λnewi = λold

i + ∆λi without molecular exchange,the condition of microscopic reversibility is given by,

P sub a Pcomp i Pα Paccαβ = P sub b Pcomp i Pβ Pacc

βα , (2.35)

where,

P sub a : probability of selecting subsystem a for an increase in the fractionalstate.

Pcomp i : probability of selecting component i for an change in the couplingparameter.

Pα : probability of the system to be in configuration α, Eq. 2.31.Paccαβ : acceptance probability for a configurational change from α to β.

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27

The probability of selecting any of both subsystems for a change in the coupling pa-rameter is the same, P sub a = P sub b = 1/2, and the probability of selecting a component iis the same for all components in the system Pcomp 1 = Pcomp 2 = ... = Pcomp i = 1/n. Thecriterion for a reversible change in λi without molecular exchange is given by,

Cαβ =Paccαβ

Paccβα

=Pβ

Pα= exp

[wi(λnew

i ) − wi(λoldi ) −

∆Ua

kBT−

∆Ub

kBT

], (2.36)

where ∆Ua and ∆Ub indicate the change in energy in both subsystems due to a change in thecoupling state of component i. It has to be noticed that for the case of chain molecules withconstant bond length considered here, the change in energy considers the intermolecularU inter as well as the intramolecular U intra contributions. A new configuration β is acceptedwith a probability given by min(1, Cαβ).

Coupling parameter change with particle transferFor a change in the coupling parameter ∆λi of component i with particle exchange (fromsubsystem a to b), the condition of microscopic reversibility is given by,

Na,i!Nb,i! P sub b Pcomp i Pmol i,aα Ppos b

α Pα Paccαβ =

(Na,i − 1)!(Nb,i + 1)! P sub b Pcomp i Pmol i,bα Ppos a

β Pβ Paccβα , (2.37)

where,

P sub b : probability of selecting subsystem b as the subsystem where the cou-pling state of the particle is increased.

Pcomp i : probability of selecting component i for a coupling state change.Pmol i,aα : probability of choosing a molecule of component i in subsystem a as

the new fractional molecule, 1/Na,i.Ppos bα : probability of choosing a position in subsystem b for the insertion of

a new fractional molecule λ = 0.Pα : probability of the system to be in configuration α, Eq. 2.31.Paccαβ : probability of accepting a change in the configuration of the system

from α to β.Pmol i,bβ : probability of choosing a molecule in subsystem b as the new frac-

tional molecule, 1/(Nb,i + 1).

The probability of the new configuration β is proportional to,

Pβ ∝ exp

n∑j,i

(ln

(N j!

Na, j!Nb, j!

)+ w j(λ j)

)+ ln

(Ni!

(Na,i − 1)!(Nb,i + 1)!

)+ wi(λi)

+(Na + n − 1) ln Va + (Nb + n + 1) ln Vb

−Ua(sNa−1

a , sna, λ1, ..., λn)

kBT−

Ub(sNb+1b , sn

b, λ1, ..., λn)kBT

. (2.38)

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28 2 Molecular simulation methods for chain fluids

Therefore, for component type i the criterion for a reversible change in λi with particleexchange is given by,

Cαβ =Paccαβ

Paccβα

=(Na,i − 1)!(Nb,i + 1)!

Na,i!Nb,i!

Pmol i,bβ

Pmol i,aα

= exp[ln

(Na,i

Nb,i + 1Vb

Va

)+ wi(λnew

i ) − wi(λoldi ) −

∆Ua

kBT−

∆Ub

kBT

], (2.39)

where ∆Ua and ∆Ub indicate the change in energy in both subsystems due to a changein the coupling state of component i with molecular transfer. As in the case of molecularexchange without particle transfer, the change in energy for chain molecules with constantbond length, the change in the energy of both subsystems considers the intermolecularU inter as well as the intramolecular U intra contributions to the energy. Here, the couplingstate of the new inserted fractional molecule is equal to λnew

i = λoldi + ∆λi − (mi + 1). The

particle transfer step (mi + 1) → 0 does not carry out any weight function change. A newconfiguration β is accepted with a probability given by min(1, Cαβ).

2.4.2. NPT expanded Gibbs ensembleThe partition function for constant pressure NPT expanded Gibbs ensemble simulations isgiven by (see section 2.3.3 for the constant pressure Gibbs ensemble),

QNPT =

(P

kBT

)2 1Λ3(N+2n)

n∏i=1

1Ni!

Ni∑Na,i=0

mi∑λi=0

Ni!Na,i!Nb,i!

exp [wi(λi)]

×

∫ ∞

0dVa exp

[−

PVa

kBT

]VNa+n

a

×

∫ ∞

0dVb exp

[−

PVb

kBT

]VNb+n

b

×

∫ 1

0dsNa

a dsna exp

−Ua(sNaa , sn

a, λ1, ..., λn)kBT

×

∫ 1

0dsNb

b dsnb exp

−Ub(sNbb , sn

b, λ1, ..., λn)kBT

. (2.40)

The constant value (P/kBT )2 is introduced to keep the partition function dimensionless [80].Volume changes are performed for each system independently, in the logarithm of the vol-ume. The criterion for accepting a trial change in the volume of each subsystem is given byEq. 2.12. The acceptance criterion for a trial change in the coupling state of a component iis given by Eq. 2.36 for a change without molecular exchange and by Eq. 2.39 for a changewith molecular exchange.

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29

2.4.3. Wang-Landau algorithmIdeally, an equal probability of visiting any fractional state is desired. For this aim, a weightfunction for every fractional state wi(λi) is introduced in the partition function (Eq. 2.30and Eq. 2.40) to bias the statistics of the system. It is clear that in the modified system,the magnitude of the weight functions has to be inversely proportional to the density ofstates of the fractional state of the non-modified system. However, the magnitude of theseweight functions cannot be know a priori and an iterative method is required for theirdetermination. Wang and Landau proposed an iterative algorithm for estimating the densityof states of systems in energy space [142, 143]. Similarly, this algorithm can be used forestimating the density of states in any other parameter space. Here, it is used for estimatingthe density of states in the coupling parameter space and therefore determining the valuesof the weight functions. The method is based on modifying the density of states of thesystem every time a fractional state is visited to produce a flat probability distribution inthe coupling parameter space (flat histogram). The density of states is changed through theweight function wi(λi) by a modification factor ν that reduces the weight of a fractional stateeach time it is visited,

wi(λi)→ wi(λi) − ν. (2.41)

When a flat histogram is obtained, the value of the modification factor ν is reduced, ν →ν/2, and the histograms are set to zero. A complete flat histogram is not possible and a flathistogram is identified when the difference between the largest and smallest frequency issmaller than 10% of the largest frequency. Changing the density of states at each step altersthe Markowian chain and only a modification factor as small as the number precision of thecomputing machine will strictly obey detailed balance [144]. This condition is practicallyimpossible and the iteration proceeds until the modification factor is reduced to a very smallvalue. A final value of 1×10−9 is considered satisfactory in this work. A starting value of1×10−5 was sufficient for reaching fast convergence avoiding large oscillations in the valueof the weight functions.

2.5 Solubility of gases2.5.1. Henry’s lawThe solubility of sligthly soluble gases in fluids is usually described using Henry’s law,which establishes a proportional relation between the partial pressure of a gas and its con-centration in the fluid solvent. The constant of proportionality is called the Henry coefficientand it is formally defined as [145],

Hk = limxk→0

fkxk, (2.42)

here fk is the fugacity of the diluted gas in the fluid mixture. The subindex k is introducedto identify the gas as the diluted solute and xk is the mole fraction of the diluted gas in thefluid.

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30 2 Molecular simulation methods for chain fluids

The fugacity of component k is related to its chemical potential by,

µk = µ0k + RT ln

fkf 0k

, (2.43)

where µk is the chemical potential of component k. The reference state is identified withthe superscript 0. A similar equation can be written for an ideal gas,

µIGk = µ0

k + RT lnfk

IG

f 0k

. (2.44)

The fugacity of component k in the ideal gas state is equal to its partial pressure, which atsame temperature T and density ρ conditions as in the non-ideal fluid is equal to, fk

IG=

xk ρkBT . Subtracting Eq 2.43 from Eq 2.44 leads to the residual chemical potential ofcomponent k [146],

µresk (T, ρ, x) = µk(T, ρ, x) − µIG

k (T, ρ, x) = kBT lnfk(T, ρ, x)xk ρkBT

, (2.45)

where x indicates the mole fraction of the fluid.The relationship between Henry coefficient and the residual chemical potential at de-

fined temperature and density is obtained by replacing Eq. 2.45 into Eq. 2.42 [146, 147],

Hk(T, ρ, x) = ρkBT exp[µres

k (T, ρ, x)kBT

]. (2.46)

This expression is obtained in the infinite dilution limit of component k, where thecomposition of the fluid x is not changed by the presence of the solute.

A dimensionless Henry coefficient that is directly related to the residual chemical po-tential is defined as following,

ln H∗k (T, ρ, x) = lnHk(T, ρ, x)ρkBT

=µres

k (T, ρ, x)kBT

. (2.47)

The relative effect of connectivity and molecular anisotropy in chain fluids is studiedby relative Henry coefficients, defined as the ratio between the Henry coefficient for thesolubility in a chain fluid Hk to the Henry coefficient for the solubility in a monomer (single-segment) fluid H0

2 at same temperature and density conditions.

Hk

H0k

= exp

µresk (T, ρ, x)

kBT−µres,0

k (T, ρ)kBT

. (2.48)

2.5.2. Widom’s test-particle insertion methodIn the test-particle insertion method proposed by Widom [148], the chemical potential of acomponent k in a fluid with mole fraction x is calculated by the energy change of a virtualinsertion of a molecule of component k in the fluid. The method is derived from the relationbetween chemical potential and the partition function of the system.

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The chemical potential of a component k in a mixture with mole fraction x at constanttemperature T and density ρ is defined by the partial molar property [149],

µk(T, ρ, x) =∂A(T, ρ, x)

∂Nk

∣∣∣∣∣T, ρ,N j,k

, (2.49)

where A(T, ρ, x) is the Helmholtz free energy which is related to the canonical partition(NVT ensemble, Eq. 1.3) as following,

A(T, ρ, x) = −kBT ln QNVT . (2.50)

The canonical partition function is defined by Eq. 1.3, which in reduced coordinates isexpressed as,

QNVT =VN

Λ3N N!

∫dsN exp

[−U(sN)kBT

]. (2.51)

In the Widom’s test particle method, a test-particle is temporary inserted in the fluid.The partition function of the actual system is identified as QNVT

N and the partition functionof the system with the inserted test-particle is denoted as QNVT

N+1 . Replacing Eq. 2.50 intoEq. 2.49, the chemical potential of component k is given by,

µk = −kBT lnQNVT

N+1

QNVTN

= −kBT ln

(V

Λ3(N + 1)

)− kBT ln

⟨exp

[−

∆U testk

kBT

]⟩= µIG + µres

k , (2.52)

where ∆U testk is the energy change in the system by the temporary insertion of a test-particle

of component k. From this last equation, it can be identified that the residual chemical po-tential of component k in the fluid is obtained from the ensemble average of the Boltzmannfactor related to the energy of insertion of the test-particle [146, 148, 150],

µresk (T, ρ, x)

kBT= − ln

⟨exp

[−

∆U test

kBT

]⟩. (2.53)

In the case of hard-sphere fluids Eq. 2.53 reduces to,

µres,HSk (T, ρ, x)

kBT= − ln〈p〉, (2.54)

where the parameter p takes the value of 1 for a virtual test-particle insertion without over-lap or 0 for a virtual insertion with overlap.

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32 2 Molecular simulation methods for chain fluids

Calculation of the residual chemical potential in the NPT ensemble is derived from thepartition function given in Eq. 2.9,

QNPT =

(P

kBT

)1

Λ3N N!

∫ ∞

0dV VN exp

[−

PVkBT

] ∫ 1

0dsN exp

[−

U(sN)kBT

]=

PkBT

∫ ∞

0dV exp

[−

PVkBT

]QNVT . (2.55)

The chemical potential of component k can be then obtained from,

µk = −kBT lnQNPT

N+1

QNPTN

= −kBT ln

(〈V〉

Λ3(N + 1)

)− kBT ln

⟨V exp

[−∆U test

k /kBT]⟩

〈V〉

= µIG + µresNPT . (2.56)

Therefore, the residual chemical potential of component k is calculated in the NPT ensem-ble from [132, 151, 152],

µres(T, ρ′, x)kBT

= − ln

⟨V exp

[−∆U test

k /kBT]⟩

〈V〉. (2.57)

It has to be noticed that in this equation calculations are performed at constant temperatureand pressure conditions and the density in Eq. 2.57 is obtained from the ensemble averageρ′ = 〈ρ〉 = 〈N/V〉. In the case of hard-sphere systems, this expression reduces to,

µres,HSk (T, ρ′, x)

kBT= − ln

〈V p〉〈V〉

, (2.58)

where the parameter p takes the value of either 1 or 0 as explained before.

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3Liquid crystal phase behavior of hard-sphere chain fluids

The liquid crystal phase behavior of linear and partially-flexible hard-sphere chain fluidsand the solubility of a hard-sphere gas in them are studied using constant pressure MonteCarlo simulations (NPT ensemble). An extensive study on the liquid crystal phase behaviorof linear hard-sphere 7-mer to 16-mer, 18-mer, and 20-mer chain fluids is shown in thischapter. The phase behavior of partially-flexible fluids with a total length of 8, 10, 14 and15 segments with different lengths in the linear part was also determined. For linear fluids,a maximum in the isotropic to nematic packing fraction change is observed for the linear15-mer and 16-mer chain fluids. The infinite dilution solubility of hard spheres in linearand partially-flexible hard-sphere chain fluids is calculated using the Widom test-particleinsertion method. To identify the effect of chain connectivity and molecular anisotropy,solubility is expressed as relative Henry coefficients, defined as the ratio between the Henrycoefficient of a gas in the chain fluid to Henry coefficient of a gas in a monomer fluidat same packing fraction. A linear relationship between relative Henry coefficients andpacking fraction is observed for all linear fluids. Furthermore, this linearity is independentof liquid crystal ordering and seems to be independent of chain length for the linear hard-sphere 10-mer and longer chains. A similar trend was observed for the solubility of hardspheres in partially-flexible fluids. At higher packing fractions, the small flexibility of thesefluids seems to improve solubility in comparison with their linear counterparts.

This chapter is based on:B. Oyarzún, T. van Westen, and T.J.H. Vlugt J. Chem. Phys. 138 (2013) 204905 [46].

33

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3

34 3 Liquid crystal phase behavior of hard-sphere chain fluids

3.1 IntroductionFluids made of linear hard-sphere chain molecules can show liquid crystal phase for chainmolecules with sufficient elongation. Vega et al. [85] studied the phase behavior of linearhard-sphere chain fluids with a length of 3, 4, 5, 6 and 7 segments (3-mer to 7-mer) fromconstant pressure Monte Carlo simulations. These authors observed that the linear hard-sphere 5-mer and longer chains form liquid crystal phases, while shorter chains experiencea direct transition from the isotropic to the crystal state [85–87]. Williamson et al. [88]investigated the phase behavior of the linear hard-sphere 7-mer fluid in all phases fromthe isotropic to the crystal state. The isotropic-nematic phase behavior of the linear hard-sphere 8-mer and 20-mer chain fluid was obtained by Yethiraj et al. [89]. Whittle et al. [90]studied the phase behavior of linear 6-mer and 8-mer fused hard-sphere chain moleculeswith length-to-width ratios ranging from 3.5 to 5.2, observing a nematic phase only for thelongest molecule.

Real liquid crystal molecules have a certain degree of flexibility which is mainly pro-vided by alkyl terminal substituents [2]. It has been shown that increasing flexibility desta-bilizes the nematic and smectic phases favoring the isotropic state [91, 153]. Introducingmolecular flexibility into hard-sphere chain molecules can be done either homogeneouslyby semi-flexible molecules where a bond-bending and torsional potential is defined forthe whole molecule, or heterogeneously by partially-flexible molecules where potentialsare defined for different parts of the molecule. The first approach was used by Wilson etal. [153, 154] for a fluid made of semi-flexible hard-sphere 7-mer chain molecules bondedby potential wells, as well as by Yethiraj et al. [89] who determined the isotropic andnematic phase behavior of semi-flexible hard-sphere 8-mer and 20-mer chain moleculeswith bending potentials of varying stiffness. A partially-flexible model was employed byMcBride et al. [91, 91, 93] for fused hard-sphere molecules composed of a linear and afreely-jointed part with a total length of 15 segments and with 8 to 15 segments in the linearpart. Escobedo et al. [94] used the expanded Gibbs ensemble and the pseudo-Gibbs ensem-ble for calculating the isotropic-nematic phase transition of hard-sphere 8-mer and 16-merchain molecules with finite and infinite bond bending potentials (linear chains).

In this chapter, simulation results are presented for the liquid crystal phase behaviorof linear and partially-flexible hard-sphere chain fluids. Simulation details are described insection 3.2. The phase behavior of linear hard-sphere chain fluids is presented in section 3.3and the effect of flexibility is studied in section 3.4 from liquid crystal phase behavior ofpartially-flexible hard-sphere chain fluids. The infinite dilution solubility of hard spheres inhard-sphere chain fluids is described in section 3.5. Results of this chapter are summarizedin section 3.6.

3.2 Simulation detailsThe phase behavior of linear and partially-flexible hard-sphere chain fluids was obtainedby Monte Carlo simulations in the isobaric-isothermal NPT ensemble. Periodic boundaryconditions were used in a rectangular simulation box with varying orthogonal dimensions.The following Monte Carlo trial moves were used: displacement, rotation, volume change,and partial-regrowth of the freely-jointed part using the Configurational-bias Monte Carlomethod [80, 100, 101, 155]. In the Configurational-bias Monte Carlo method 10 trail direc-

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3.2 Simulation details

3

35

tions were attempted for regrowing each segment in the freely-jointed part. A system size ofN=350 molecules was used for all simulations except for the linear hard-sphere 7-mer and8-mer chain fluids in which a larger number of 576 molecules was employed. The latter wasdone in order to compare actual simulation results with previous simulation data, especiallywith that obtained by Williamson et al. [88] for the linear hard-sphere 7-mer chain fluid.Simulations with 216 and 350 molecules were also performed for the linear 7-mer in orderto study the system size effect. In the equilibration period, the maximum displacement androtation angle were adjusted in order to obtain an acceptance ratio of 20%. Rotation aroundevery Cartesian axes was considered separately due to the orientation of molecules aroundthe nematic director in nematic and smectic phases. Volume changes were performed in ananisotropic manner by choosing independently the box side which is expanded/contracted[89, 91]. In this way, the natural anisotropy introduced by a rectangular box is decoupledfrom the phase anisotropy inherent to ordered systems. Displacement and rotation trialmoves were selected with the same probability, while the probability of attempting a vol-ume change was set to 2%. In the case of partially-flexible molecules, 15% of the trialmoves were attempted to regrow the freely-jointed part. The anisotropic expansion of thesimulation box may result in the contraction of one of its dimensions to zero due to thefinite size of the system. This event is avoided by using large system sizes but at the costof long simulation runs. A good balance between a very low occurrence of this event andshort simulation runs was obtained for a system size of 350 molecules. A typical simulationrun comprises 1×106 Monte Carlo cycles in the equilibration period and 1×106 cycles inthe production period. Every Monte Carlo cycle consisted of a total of N Monte Carlo trialmoves. However, close to the isotropic-nematic transition region, the required number ofsimulation cycles in the equilibration and production periods was normally doubled due tolarge fluctuations in the observed variables.

Initial isotropic and nematic configurations were used for the simulation of linear hard-sphere chain fluids. For partially-flexible systems, however, only nematic initial configura-tions were used, since for these systems simulations starting from an isotropic configurationrequired very long equilibration periods, typically larger than 5×106 Monte Carlo cycles.For linear chains, simulation starting from an isotropic configuration were obtained for aregion close to the isotropic-nematic phase transition in order to verify that results in theisotropic and nematic phases are independent of the initial configuration, and to determinethe hysteresis at the isotropic-nematic phase transition. Isotropic initial configurations wereobtained by placing molecules in the simulation box randomly without a preferred orien-tation. A nematic configuration was generated by placing molecules in random positionsbut perfectly aligned on the direction of the largest dimension of the simulation box. Inthe case of partially-flexible molecules, an initial linear molecular configuration was cho-sen. The initial dimensions of the simulation box were selected in order to ensure that theinitial packing fraction of the system is located in either the isotropic or nematic regions.Every simulation run was started from an independent initial configuration in order to avoidcorrelation between results.

Simulations were performed at constant reduced pressure defined as P∗ = P vmol/kBT ,where P is pressure, kB is the Boltzmann constant, T is temperature, and vmol = mπd3/6is the volume of a molecule made of m beads with diameter d set to unity. The densityis expressed dimensionless by the packing fraction η = Nvmol/V , where N is the number

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36 3 Liquid crystal phase behavior of hard-sphere chain fluids

0

2

4

6

8

0.2 0.3 0.4

P*

η

(a)

0

0.2

0.4

0.6

0.8

1

0.2 0.3 0.4

S2

η

(b)

Figure 3.1: Liquid crystal phase behavior of the linear hard-sphere 7-mer chain fluid: (a) reduced pressure P∗,(b) order parameter S 2 as a function of packing fraction η. Results obtained from initial isotropic and nematicconfigurations are denoted by closed (•) and open circles (◦) respectively. Previous data from Williamsonand Jackson [88] are denoted by crosses (×) for initial isotropic configurations and by pluses (+) for initialcrystal configurations. In addition, results of Vega et al. [85] started from a crystal structure are indicated byan asterisk (∗). The size of the symbols is larger than the relative error in the packing fraction (1-2%) and therelative error in the order parameter (2-5%) .

of molecules. The orientational order of the system is measured by the order parameter S 2Eq. 2.3, which has a value of 0 in an isotropic fluid and a value of 1 for a completely alignednematic phase.

The solubility of hard spheres in hard-sphere chain fluids at infinite dilution is describedby Henry coefficients defined by Eq. 2.46. The residual chemical potential is calculated bythe Widom test-particle insertion method [148], which for Monte Carlo simulations of hardspheres at constant pressure takes the form of Eq. 2.54 [152]. A total of 100 test-particleinsertions are performed for each configuration sampled every 1×103 Monte Carlo cycles.

3.3 Linear hard-sphere chain fluidsA systematic study on the liquid crystal phase behavior was performed for linear 7-mer to16-mer, 18-mer, and 20-mer hard-sphere chain fluids. Previous simulation data for the lin-ear 7-mer [85, 88], 8-mer and 20-mer [89] are present in literature. Compared to previousstudies, simulation runs in this work were significantly longer (typically by a factor of 2 to10) in particular close to the isotropic-nematic phase transition. In addition, a large num-ber of simulations was performed close to the isotropic-nematic phase transition in order todescribe the transition region with high accuracy. The phase behavior of the 7-mer was al-ready determined by Williamson et al. [88] and Vega et al. [85]. Due to the large collectionof data existing for this system, the linear hard-sphere 7-mer molecule can be consideredas a suitable reference system. Nevertheless, new and more precise simulation data for the7-mer was obtained. Fig. 3.1 shows the liquid crystal phase behavior of the linear hard-sphere 7-mer chain fluid compared to literature data. It can be observed that all simulationdata are comparable with slight differences at reduced pressures of 5.0 and higher. From theextensive results obtained in this work the isotropic-nematic phase transition is found at a

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3.3 Linear hard-sphere chain fluids

3

37

0

1

2

3

4

5

6

0.2 0.3 0.4

P*

η

(a)

0

0.04

0.08

0 0.04 0.08

S2

1/√N

0

0.2

0.4

0.6

0.8

1

0.2 0.3 0.4

S2

η

(b)

Figure 3.2: Finite size effect on: (a) reduced pressure P∗, (b) order parameter S 2 for the linear hard-sphere 7-mer chain fluid as a function of packing fraction η. All results were obtained from initial nematic configurationswith system sizes of 216 (4), 350 (�) and 576 (◦) molecules. The inset in figure (b) shows the system sizedependence of the order parameter for the linear 7-mer in the isotropic region at a reduced pressure ofP∗=1 and at a packing fraction η∼0.178 for systems sizes of 216, 350 and 576 molecules. In this sub-figure,values are the average of four independent simulation results and the symbol size is larger than one standarddeviation. The solid line is a linear fit with R2 = 0.999.

reduced pressure of P∗∼3.39−3.49 at a packing fraction of ηI∼0.289−0.291 for the isotropicphase and of ηN∼0.302−0.303 for the nematic phase. These values lie in between the rangesP∗∼3.15−3.78, ηI∼0.266−0.303 and ηN∼0.285−0.312 reported by Williamson et al. [88],and it is in agreement with the reduced pressure range P∗∼2.9−3.7 and packing fractionsηI≈0.274 and ηN≈0.308 reported by Vega et al. [85]. The large hysteresis at the isotropic-nematic phase transition observed by Williamson et al. [88] was not detected here. It issuspected that close to the isotropic-nematic phase transition, where fluctuations are large,the simulations results of Williamson and Jackson may either not be well equilibrated ortheir sampling period was too short (4×105 compared to 2×106 of this work). Nevertheless,a narrow hysteresis of 0.1 in reduced pressure was detected for the linear 7-mer. A phasetransition to the smectic-A phase was observed at a reduced pressure P∗∼4.80−5.00 andηN≈0.359, ηS≈0.380.

The system size effect of the linear 7-mer was also studied and results for systems sizesof 216, 350 and 576 molecules are compared in Fig. 3.2. It can be observed that results forreduced pressure versus packing fraction at the isotropic and nematic regions for all threesystems are comparable. However, the precise location of the nematic-smectic phase tran-sition seems to be system size dependent. For the larger 576 molecules system, a first-orderphase transition to a smectic-A phase is detected at a reduced pressure between 4.8 and 5.0.On the contrary, for the smaller 350 and 216 molecules systems, partial smectic orderingwith no clear defined smectic layers was observed for lower reduced pressure between 4.0and 5.0. For all three system sizes, smectic-A layers were clearly recognized only at re-duced pressures of 5.0 and higher. This system size dependence on the nematic-smecticphase transition was already pointed out by Polson et al. [98] for the nematic to smecticphase transition of hard spherocylinders in the limit of infinite aspect ratio. In their study,the free-energy profile at the nematic-smectic phase transition was calculated for different

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38 3 Liquid crystal phase behavior of hard-sphere chain fluids

system sizes (928, 540 and 280 molecules) as a function of smectic ordering. These au-thors found that a large Gibbs energy barrier between the nematic and the smectic stateexists only for the larger system. The Gibbs energy barrier was negligible for the interme-diate system size and vanishing for the smaller one. In their work, Polson et al. concludedthat, in the thermodynamic limit, the nematic to smectic phase transition of spherocylindersseems to be of first-order for all aspect ratios. Similarly, here, a first order nematic-smecticphase transition was observed only for the larger system size of 576 molecules and the par-tial smectic ordering observed in the smaller systems can be a consequence of a low Gibbsenergy barrier caused by the reduced system size.

Another effect of the finite size of the system can be identified in Fig. 3.2 from the orderparameter in the isotropic region. It can be observed that in the isotropic region all threesystems show a positive value different than zero for the order parameter, approaching asystem-size dependent limiting value as the packing fraction is reduced. The origin ofthis finite order in the isotropic phase and its corresponding finite order parameter valuewas described by Eppenga et al. [34]. For isotropic systems of finite size, these authorsdemonstrated the largest eigenvalue of the Q-tensor (from which the order parameter isestimated) deviates from zero by 1/

√N. The inset in Fig. 3.2 (b) shows the relation between

order parameter values and system size at same packing fraction. It can be observed thatthe 1/

√N dependence is almost exact leading to a vanishing value of the order parameter

on the thermodynamic limit.The phase behavior of the linear hard-sphere 8-mer to 16-mer, 18-mer, and 20-mer

chain fluids is shown in Figs. 3.3-3.5. Literature data for the linear 8-mer and 20-mer fromYethiraj et al. [89] are also included. These authors used a slightly different definition forthe order parameter, based on the middle eigenvalue of the Q-tensor instead of the largestone used here. The middle eigenvalue leads to values of the order parameter that fluctu-ate around zero in the isotropic phase. Although both definitions are directly related (asshown in the previous section), the definition based on the largest eigenvalue is consideredto be more consistent with the mathematical framework of the Q-tensor. Initial isotropicand nematic configurations led to hysteresis in the isotropic-nematic phase transition. Themagnitude of the hysteresis showed to be dependent of the packing fraction difference atthe isotropic-nematic transition, varying from a difference in reduced pressure of ∼0.04 forthe linear 7-mer to ∼0.17 for the linear 15-mer fluid.

Fig. 3.2 and Table 3.1 summarize the reduced pressure and packing fraction change atthe isotropic-nematic transition for all linear systems studied. Reduced transition pressureswere calculated at the middle-point of the hysteresis, which was estimated by the average ofthe reduced pressure at coexistence for results starting from an initial isotropic and nematicconfigurations. Packing fractions were obtained by fitting a third order polynomial on thepacking fraction as a function of reduced pressure for each isotropic and nematic branchesseparately. The isotropic and nematic packing fraction at phase transition were obtainedusing the previous calculated reduced transition pressures in the fitted equations. Fig. 3.2also includes the available literature data for the isotropic-nematic phase transition of linearhard-sphere chain fluids. It is observed that the literature data is comparable with the resultsfor the reduced pressure at coexistence but no systematic behavior can be identified fromliterature data for the packing fraction difference. From Fig. 3.2 (a), it can be observedthat with increasing chain length, the transition pressure decreases in a continuous manner.

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3.3 Linear hard-sphere chain fluids

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39

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3

S2

η

(b)

8

0 0.1 0.2 0.3

9

0 0.1 0.2 0.3

10

0 0.1 0.2 0.3 0.4

11

0

1

2

3

4

5

0 0.1 0.2 0.3

P*

η

(a)

8

0 0.1 0.2 0.3

9

0 0.1 0.2 0.3

10

0 0.1 0.2 0.3 0.4

11

Figure 3.3: Liquid crystal phase behavior of linear hard-sphere 8-mer to 11-mer chain fluids: (a) reducedpressure P∗, (b) order parameter S 2 as a function of packing fraction η. Results obtained from initial isotropicand nematic configurations are denoted by closed (•) and open circles (◦) respectively. Previous simulationdata from Yethiraj et al. [89] for the 8 linear chain are indicated by an asterisk (∗).

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40 3 Liquid crystal phase behavior of hard-sphere chain fluids

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3

S2

η

(b)

12

0 0.1 0.2 0.3

13

0 0.1 0.2 0.3

14

0 0.1 0.2 0.3 0.4

15

0

1

2

3

4

5

0 0.1 0.2 0.3

P*

η

(a)

12

0 0.1 0.2 0.3

13

0 0.1 0.2 0.3

14

0 0.1 0.2 0.3 0.4

15

Figure 3.4: Liquid crystal phase behavior of linear hard-sphere 12-mer to 15-mer chain fluids: (a) reducedpressure P∗, (b) order parameter S 2 as a function of packing fraction η. Results obtained from initial isotropicand nematic configurations are denoted by closed (•) and open circles (◦) respectively.

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3.3 Linear hard-sphere chain fluids

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41

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3

S2

η

(b)

16

0 0.1 0.2 0.3

18

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2

20

0

1

2

3

4

5

0 0.1 0.2 0.3

P*

η

(a)

16

0 0.1 0.2 0.3

18

0

0.5

1

1.5

0 0.1 0.2

20

Figure 3.5: Liquid crystal phase behavior of linear hard-sphere 16-mer, 18-mer, and 20-mer chain fluids" (a)reduced pressure P∗, (b) order parameter S 2 as a function of packing fraction η. Results obtained from initialisotropic and nematic configurations are denoted by closed (•) and open circles (◦) respectively. Previoussimulation data from Yethiraj et al. [89] are indicated by an asterisk (∗).

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42 3 Liquid crystal phase behavior of hard-sphere chain fluids

Table 3.1: Reduced pressure P∗, packing fraction of the isotropic phase ηI , packing fraction of the nematicphase ηN and packing fraction change at the isotropic-nematic phase transition ∆ηN−I for linear moleculeswith a length of linear hard-sphere 7-mer to 16-mer, 18-mer, and 20-mer chain fluids.

m P∗ ηI ηN ∆ηN−I

7 3.47 0.2941 0.3009 0.00688 2.65 0.2560 0.2678 0.01189 2.08 0.2269 0.2415 0.014610 1.76 0.2065 0.2231 0.016611 1.53 0.1900 0.2083 0.018312 1.32 0.1733 0.1935 0.020213 1.20 0.1625 0.1838 0.021314 1.09 0.1528 0.1747 0.021915 0.99 0.1429 0.1649 0.022016 0.94 0.1388 0.1610 0.022018 0.78 0.1207 0.1425 0.021820 0.65 0.1073 0.1250 0.0177

0

1

2

3

4

4 6 8 10 12 14 16 18 20 22

P*

m

(a)

0

0.005

0.01

0.015

0.02

0.025

4 6 8 10 12 14 16 18 20 22

∆ηN-I

m

(b)

Table 3.2: (a) Transition pressures P∗, and (b) packing fraction change ∆ηN−I at the isotropic-nematic co-existence for linear hard-sphere 8-mer to 16-mer, 18-mer, and 20-mer chain fluids. Previous data fromWilliamson et al.[88] for the linear 7-mer (+), Yethiraj et al. [89] for the linear 8-mer and 20-mer (×) andfrom Escobedo and et al. [94] for the linear 8-mer and 16-mer (∗) are also indicated.

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43

Table 3.3: Reduced pressure P∗, packing fraction of the isotropic phase ηI , packing fraction of the nematicphase ηN and packing fraction change at the isotropic-nematic phase transition ηN−I for partially-flexiblehard-sphere 15-14-mer, 15-13-mer, 15-12-mer, 14-13-mer and 14-12-mer chain fluids.

m − mR P∗ ηI ηN ∆ηN−I

15-14 1.28 0.1577 0.1740 0.016315-13 1.77 0.1788 0.1906 0.011815-12 2.54 0.2071 0.2174 0.010314-13 1.44 0.1689 0.1841 0.015214-12 2.12 0.2034 0.2139 0.0105

On the other hand, from Fig. 3.2 (b) it can be observed that the packing fraction differenceshows a maximum for the linear 15-mer and the 16-mer. This is a rather surprising resultthat has not been explicitly reported in the simulation of hard-sphere chain systems before.Bolhuis et al. [44] obtained for spherocylinders the isotropic and nematic densities at co-existence as a function of shape anisotropy L/D by a modified Gibbs-Duhem integrationmethod. From their data, it can be identified that a maximum is reached in the isotropic-nematic density difference at coexistence for a shape anisotropy between 15 and 20. Froman Onsager type density functional theory, Fynewever et al. [156] show a similar behaviorof the packing fraction difference for linear hard-sphere chain fluids as a function of chainlength, with a maximum at a chain length close to 10. However, neither of these studiesmentions the existence of a maximum in the packing fraction difference explicitly.

3.4 Partially flexible hard-sphere chain fluidsThe liquid crystal phase behavior of partially-flexible molecules is investigated for differentmolecular lengths with varying degree of flexibility. Results for this section were obtainedfrom an initial nematic configuration only, since simulations started from an initial isotropicconfiguration required excessively long equilibration periods. In Fig. 3.6, the phase behav-ior of the partially-flexible hard-sphere 15-12-mer, 15-13-mer, and 15-14-mer chain fluidsis shown. It can be observed that increasing flexibility destabilizes the nematic phase withrespect to the isotropic phase, leading to higher transition pressures at higher coexistencepacking fractions. The isotropic to nematic phase transition is driven by a difference in theorientational average of the excluded volume between the isotropic and nematic phase [20].Flexibility reduces this difference [157], therefore a higher pressure is required in order toinduce the transition to the nematic state. Reduced pressures and packing fraction changesat the isotropic-nematic phase transition are summarized in Table 3.3. It can be identifiedthat flexibility increases the transition pressure and reduces the isotropic-nematic packingfraction difference. A transition to a smectic-A phase is identified at a reduced pressure inthe range of 3.0 − 3.5 for the partially-flexible hard-sphere 15-14-mer and 15-13-mer chainfluids and between 3.40 − 3.60 for the partially-flexible 15-12-mer.

The average end-to-end length in the isotropic and nematic regions for the partially-flexible hard-sphere 15-14-mer, 15-13-mer and 15-12-mer chain fluids is shown in Fig. 3.7as a function of pressure. It can be clearly identified that in the isotropic region the averageend-to-end length decreases as pressure increases. This behavior is a consequence of the

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44 3 Liquid crystal phase behavior of hard-sphere chain fluids

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3

S2

η

(b)

15-12

0 0.1 0.2 0.3

15-13

0 0.1 0.2 0.3 0.4

15-14

0

1

2

3

4

5

0 0.1 0.2 0.3

P*

η

(a)

15-12

0 0.1 0.2 0.3

15-13

0 0.1 0.2 0.3 0.4

15-14

Figure 3.6: Liquid crystal phase behavior of partially-flexible hard-sphere 15-12-mer, 15-13-mer and 15-14-mer chain fluids: (a) reduced pressure P∗, (b) order parameter S 2 as a function of packing fraction η.

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3.4 Partially flexible hard-sphere chain fluids

3

45

0.82

0.83

0.84

0.85

0 1 2 3 4 5

<R>/m

P*

15-12

0.88

0.89

0.9

0.91

0 1 2 3 4 5

15-13

0.95

0.96

0 1 2 3 4 5

15-14

Figure 3.7: Average end-to-end length of partially-flexible hard-sphere 15-14-mer, 15-13-mer, 15-12-merchain fluids. Results are shown relative to the total chain length 15-mer.

tendency of molecules to be more entangled at increased packing fractions. On the contrary,a step increase in the average end-to-end length occurs at the isotropic to nematic phasetransition. This increased molecular elongation is a result of the orientational ordering inthe nematic phase, forcing molecules to stretch along the phase director. Nevertheless, thisstep-difference in the average end-to-end length at coexistence is relatively small, varyingfrom ∼0.2% for the partially-flexible 15-14-mer to ∼1% for the partially-flexible 15-12-mer.In the nematic region, the average end-to-end length increases asymptotically towards thetotal molecular length with increasing pressure. Although changes in the average end-to-end length are relatively small, Fig. 3.7 shows that the average molecular configuration ofpartially-flexible hard-sphere molecules varies with reduced pressure (packing fraction) andwith molecular ordering, either isotropic and nematic. This change in the average molecularconfiguration may have an effect on the free volume of the fluid [157], influencing thesolubility of hard spheres in hard-sphere fluids as it will be discussed in section 3.5.

Fig. 3.8 shows the liquid crystal phase behavior of the partially-flexible hard-sphere14-13-mer, 14-12-mer and 14-10-mer chain fluids. A nematic phase is identified only forthe partially-flexible 14-13-mer and 14-12-mer. Reduced pressures and packing fractionchanges at the isotropic-nematic phase transition for these fluids are specified in Table 3.3.A transition to the smectic-A phase is identified at a reduced pressure in the range of 3.0−3.5for the 14-13 fluid and 3.5 − 4.0 for the 14-12 fluid. The more flexible partially-flexible14-10-mer does not show a nematic phase, experiencing a continuous transition from theisotropic to the smectic-A phase. The fact that for the partially-flexible 14-10-mer theisotropic to smectic phase transition is continuous is possibly a consequence of a negligibleenergy barrier between these phases due to the reduced system size as was already pointedout in section 3.3 for the linear hard-sphere 7-mer chain fluid. However, this premise wasnot tested here and simulations in larger systems would be required. A clear smectic-Aphase could not be recognized for the partially-flexible hard-sphere 14-10-mer chain fluid,however partial smectic-A ordering was observed for reduced pressures of 5.0 and 5.2. Aclear smectic-C phase is observed at a reduced pressure of 8.0 but partial smectic-C layering

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46 3 Liquid crystal phase behavior of hard-sphere chain fluids

0

0.2

0.4

0.6

0.8

1

0.1 0.2 0.3

S2

η

(b)

14-10

0.1 0.2 0.3

14-12

0 0.1 0.2 0.3 0.4

14-13

0

2

4

6

8

10

0.1 0.2 0.3

P*

η

(a)

14-10

0.1 0.2 0.3

14-12

0 0.1 0.2 0.3 0.4

14-13

Figure 3.8: Liquid crystal phase behavior of partially-flexible hard-sphere 14-10-mer, 14-12-mer and 14-13-mer chain fluids: (a) reduced pressure P∗, (b) order parameter S 2 as a function of packing fraction η.

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3.5 Solubility of hard spheres in hard-sphere chain fluids

3

47

is already identified at reduced pressures between 5.4 and 7.0.Fig. 3.9 shows the phase behavior of the partially-flexible hard-sphere 8-7-mer, 8-6-

mer and 10-8-mer chain fluids. A continuous isotropic to smectic-A phase transition isobserved for the partially-flexible 8-7-mer system in the reduced pressure range of 5.4−5.8and clear smectic-A layers are identified at reduced pressures of 5.9 and higher. In thepartially-flexible 8-6-mer system, clearly defined smectic-A layers could not be identified,however partial smectic ordering was observed at a reduced pressure of 14 and higher. Forthis system, a continuous phase transition from the isotropic to the smectic-A state wasobserved between reduced pressures of 9.4 and 10.4. Finally, the partially-flexible 10-8-mer shows a continuous phase transition from the isotropic to the smectic-A state betweenpressures of 4.7 and 5.6 and clear smectic-A layers at pressures of 5.8 and higher.

3.5 Solubility of hard spheres in hard-sphere chain fluidsThe solubility of hard spheres in a hard-sphere chain fluid depends on the available freevolume in the fluid, which is in principle determined by the molecular configuration, pack-ing fraction, and orientation of all molecules in the fluid [157, 158]. Here, the relativeimportance of molecular configuration, i.e. connectivity and rigidity ,is investigated bymeans of relative Henry coefficients defined by Eq. 2.48. The residual chemical potentialof a hard sphere in the monomer fluid, i.e. the hard-sphere fluid, H0

k is calculated from theBoublik-Mansoori-Carnahan-Starling-Leland equation of state [159–161]. The validity ofthis equation was verified with simulation results, showing an excellent agreement for theresidual chemical potential of hard spheres in a hard sphere fluid.

Fig. 3.10 (a) shows relative Henry coefficients for the solubility of hard spheres in linearhard sphere chain fluids and Fig. 3.10 (b) shows the relative Henry coefficients for thesolubility of hard spheres in partially-flexible hard sphere chain fluids. From Fig. 3.10(a), a linear relationship between relative Henry coefficients and packing fraction can beidentified for all linear fluids. A linear fit calculated from the results for linear hard-sphere10-mer and longer chain fluids is included in Fig. 3.10 (a) and (b). This linearity seems tobe independent of liquid crystal ordering and fairly independent of chain length for linearhard-sphere 10-mer and longer chain fluids.

In contrast to linear hard-sphere chain molecules, the average molecular configurationof partially-flexible hard-sphere molecules changes with packing fraction as it was shownin Fig. 3.7. It can be observed that at packing fractions lower than η = 0.23 the solubil-ity of hard spheres in 15-14, 15-13, 15-12, 14-13 and 14-12 partially-flexible hard-spherechain fluids follows the same linear relationship with packing fraction as for the linear hard-sphere chain fluids. As it is shown in Table 3.3, the isotropic-nematic phase transition of thenematic forming partially-flexible hard-sphere chain fluids takes place at packing fractionsnot higher than 0.2174, indicating that partial flexibility do not affect the relative behaviorof solubility with packing fraction at the isotropic-nematic phase transition. The effect oflarger flexibilities is also shown in Fig. 3.10 (b), which includes relative Henry coefficientscalculated from literature data for the chemical potential of hard spheres in a freely-jointedhard-sphere 16-mer chain fluid [119, 163]. It can be observed that, at low packing frac-tions η < 0.23, relative Henry coefficients for the partially-flexible and fully flexible chainmolecules do not appreciably deviate from the linear behavior derived from the linear hard-sphere chain fluids. At packing fractions higher than η=0.23, the solubility of hard spheres

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48 3 Liquid crystal phase behavior of hard-sphere chain fluids

0

0.2

0.4

0.6

0.8

1

0.1 0.2 0.3 0.4

S2

η

(b)

10-8

0.1 0.2 0.3 0.4

8-6

0.1 0.2 0.3 0.4 0.5 0.6

8-7

0

2

4

6

8

10

0.1 0.2 0.3 0.4

P*

η

(a)

10-8

0.1 0.2 0.3 0.4

8-6

0.1 0.2 0.3 0.4 0.5 0.6

8-7

Figure 3.9: Liquid crystal phase behavior of partially-flexible hard-sphere 10-8-mer, 8-6-mer and 8-7-merchain fluids: (a) reduced pressure P∗, (b) order parameter S 2 as a function of packing fraction η.

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3.5 Solubility of hard spheres in hard-sphere chain fluids

3

49

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

η

Hk

Hk

0

___

(a)

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

η

Hk

Hk

0

___

(a)

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

η

Hk

Hk

0

___

(b)

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

η

Hk

Hk

0

___

(b)

Figure 3.10: Relative Henry coefficients for the solubility of hard spheres in: (a) linear hard-sphere 7-mer (�),9-mer (�), 10-mer (◦), 11-mer (•), 12-mer (4), 13-mer (N), 14-mer (O), 15-mer (H) and 20-mer (�); (b) partially-flexible hard-sphere chain fluids 15-14-mer (�), 15-13 (�), 15-12 (◦), 14-13 (•) and 14-12 (4). The solid line isa linear fit obtained with the data of the linear hard-sphere 10-mer and longer chain fluids, with slope -2.580and intersection 1.036 (R2=0.991). Relative Henry coefficients for hard spheres in a hard dumbbell fluid (×)were calculated from literature data for the chemical potential of hard spheres in a hard dumbbell fluid byBen-Amotz and Omelyan [162]. Relative Henry coefficients for hard spheres in a freely-jointed hard-sphere16-mer chain fluid were calculated from the data of Escobedo and de Pablo [119] (∗) and from the data ofKumar et al. [163] (×).

seems to increase for the partially-flexible hard-sphere chain fluids and to decrease for thefreely-jointed hard-sphere chain fluid at the same packing fraction. A more clear insight tothis effect is given in Fig. 3.11, where the effect of molecular flexibility in packing fractionis shown by the dimensionless rigidity parameter defined as the ratio between the numberof rigid bonds angles to the total bond angles in the molecule [77, 157],

χR =

mR − 2m − 2

; m > 2

1 ; m ≤ 2. (3.1)

Fig. 3.11 (a) shows that at low packing fractions flexibility χR→0 effectively increasesthe packing fraction of the fluid, reducing in principle the solubility of gases in the liquidcrystal phase. However, this increment is relatively small (∼4%), which explains togetherwith the low values for the packing fraction the unnoticeable difference of relative Henrycoefficients between linear and fully flexible chain fluids at low packing fractions. At highpacking fractions, Fig. 3.11 (b), a minimum in packing fraction is observed as flexibilityincreases. For packing fractions below the value corresponding to χR=0, the same packingfraction value can be related to two different values of the rigidity parameter, one closer tothe linear rigid chain χR=1 and one closer to the flexible chain χR=0. The same argumentcan be used for the behavior of solubility with flexibility at high packing fractions, i.e. twosolubilities regimes, one closer to the linear chain fluid and one similar to the fully-flexiblechain. For chain fluids with high flexibility, increasing rigidity reduces the packing fractionof the fluid with the consequence of a higher solubility compared to the fully-flexible case.

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3

50 3 Liquid crystal phase behavior of hard-sphere chain fluids

0.095

0.0975

0.1

0.1025

0.105

0 0.2 0.4 0.6 0.8 1

η

χR

(a)

0.25

0.26

0.27

0 0.2 0.4 0.6 0.8 1

η

χR

(b)

Figure 3.11: The effect of molecular rigidity χR on the packing fraction η of a hard-sphere 8-mer chain fluid atconstant pressure: (a) P∗ = 0.3, (b) P∗ = 2.7.

For chain fluids with high rigidity, increasing flexibility reduces also the packing fractionof the fluid with the consequence of a higher solubility compared to the linear case. Thisbehavior explains in principle the observed increase in solubility due to flexibility for thepartially-flexible molecules shown in Fig. 3.10 (b).

To verify the independence of solubility with nematic ordering, constant volume MonteCarlo simulations at a packing fraction in-between the isotropic-nematic phase transitionwere performed using an umbrella sampling technique for the order parameter [80]. Inthese simulations, order parameter values were restricted to ranges of width 0.05 between0 and 1. The simulation of every order parameter range was performed separately from aninitial nematic configuration, which was allowed to freely evolve until a configuration withan order parameter value inside the desired range was obtained. From there on, trial movesthat try to change the phase configuration to a new one with an order parameter outside theselected order parameter range were rejected. Fig. 3.12 shows the results for relative Henrycoefficients as a function of order parameter for a linear hard-sphere 7-mer and 15-merchain fluid at a constant packing fraction in-between the isotropic-nematic phase transition.From this figure, it can be clearly identified that relative Henry coefficients are independentof the nematic ordering, being a constant for a determined value of the packing fraction.This result indicates that at constant packing fraction the orientation of molecules does notinfluence the free volume between a hard sphere and a linear hard-sphere chain molecule.Although in the low packing fraction limit this result is expected due to a negligible inter-action between fluid molecules, at finite values of the packing fraction this behavior couldnot be known a priori due to multi-body interactions in the fluid.

3.6 ConclusionsAn extended description of the liquid crystal phase behavior of linear hard-sphere 7-mer to16-mer, 18-mer and 20-mer chain fluids was obtained from constant pressure Monte Carlosimulations. For these systems, an accurate description of the isotropic to nematic phasetransition was obtained, showing a decrease in the reduced transition pressure with increas-

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3.6 Conclusions

3

51

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1

S2

Hk

Hk

0

___

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1

S2

Hk

Hk

0

___

Figure 3.12: Relative Henry coefficients as a function of order parameter S 2 for the solubility of hard spheresin a linear hard-sphere 7-mer chain fluid at constant packing fraction η = 0.295 (�) and in a linear hard-sphere15-mer chain fluid at constant packing fraction η = 0.147 (4).

ing chain length and a maximum in the isotropic to nematic packing fraction change for thelinear 15-mer and 16-mer. The liquid crystal phase behavior of the partially-flexible hard-sphere 15-14-mer, 15-13-mer, 15-12-mer, 14-13-mer, 14-12-mer, 14-10-mer, 10-8-mer, 8-7-mer and 8-6-mer chain fluids was also determined. A nematic phase was only observedfor the partially-flexible 15-14-mer, 15-13-mer, 15-12-mer, 14-13-mer, 14-12-mer chainfluids. Other partially-flexible fluids experienced a direct transition from the isotropic to thesmectic state. Flexibility increases the transition pressures reducing the isotropic-nematicpacking fraction difference. The infinite dilution solubility of hard spheres in linear andpartially-flexible hard-sphere chain fluids was also studied. A linear relationship betweenrelative Henry coefficients and packing fraction was observed for all linear hard-spherechain fluids. Moreover, this linearity seems to be independent of chain length for the linear10-mer and longer linear chains. The influence of molecular ordering on solubility was alsoinvestigated, showing that at constant packing fraction the nematic ordering does not influ-ence the solubility of hard spheres in liquid crystal fluids. Results for the solubility of hardspheres in the nematic forming partially-flexible hard-sphere 15-14-mer, 15-13-mer, 15-12-mer, 14-13-mer and 14-12-mer chain fluids showed that in the isotropic-nematic transitionregion neither ordering or flexibility affected the linear behavior between relative Henrycoefficients and packing fraction obtained from the linear fluids. However, it was observedthat at higher packing fractions the solubility of hard spheres seems to increase for the lowflexibilities of the nematic forming partially-flexible hard-sphere chain fluids and decreasefor a freely-jointed hard-sphere chain fluid.

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4Isotropic-nematic phase equilibrium of hard-sphere chainfluids

The isotropic-nematic phase equilibria of single component linear hard-sphere chain fluidsand binary mixtures of them are obtained from Monte Carlo simulations. In addition, theinfinite dilution solubility of hard spheres in the coexisting isotropic and nematic phases isdetermined from the residual chemical potential of hard spheres in the fluid. Phase equi-libria calculations are performed in an expanded formulation of the Gibbs ensemble. Thismethod allows to carry out an extensive simulation study on the phase equilibria of purelinear hard-sphere 7-mer to 20-mer chain fluids and binary mixtures of an 8-mer with a14-mer, a 16-mer and a 19-mer. The effect of molecular flexibility on the isotropic-nematicphase equilibria is assessed on the 8-mer+19-mer mixture by allowing one and two freely-jointed segments at the end of the longest molecule. Results for binary mixtures are com-pared with the theoretical predictions of van Westen et al. Ref. 78. Excellent agreementbetween theory and simulations is observed. The infinite dilution solubility of hard spheresin the hard-sphere fluids is obtained by the Widom test-particle insertion method. As in theprevious chapter on single-component hard-sphere chains, a linear relationship betweenrelative infinite dilution solubility (relative to that of hard spheres in a hard-sphere fluid)and packing fraction is found. It is observed that the solubility difference between coex-isting isotropic and nematic phases is enhanced in binary mixtures compared to the purecomponents.

This chapter is based on:B. Oyarzún, T. van Westen, and T.J.H. Vlugt J. Chem. Phys. 142 (2015) 064903 [164].

53

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4

54 4 Isotropic-nematic phase equilibrium of hard-sphere chain fluids

4.1 IntroductionLiquid crystal phases have been observed in molecular simulation studies of anisotropichard molecules with diverse shapes [33–45]. From all possible anisotropic systems, tan-gent hard-sphere chains [46, 85, 88, 89] are the most simple segment-based model of liquidcrystals. Segment-based molecules are relevant in the development of physically-based per-turbation theories describing the behavior of liquid crystals [78, 83]. Linear and partially-flexible hard-sphere chains formed by a linear part and a freely-jointed part are used forstudying the properties of liquid crystal fluids. Partial flexibility is introduced to reproducethe experimental observation that a certain degree of flexibility is required for the stabilityof liquid crystal phases over the crystal state [2, 165]. Flexibility is introduced to studyits effect on the isotropic-nematic phase transition. Molecular simulation results for thesesystems are scarce, principally due to the difficulties on performing phase equilibria calcu-lations of non-simple fluids with classical simulation techniques. In this work, an extendedmolecular simulation study on the isotropic-nematic phase equilibria of hard-sphere chainfluids is carried out. Furthermore, simulation results are compared with the theoreticalpredictions obtained from a recently developed equation of state [77, 78, 83].

In this chapter, simulation results for the isotropic-nematic phase transition of single-component and binary mixtures of hard-sphere chain fluids are presented. Details of theemployed simulation technique are described in section 4.2. Simulation results for theisotropic-nematic phase equilibria of single-component linear hard-sphere chain fluids arereported in section 4.3. Simulation results for binary mixtures of linear hard-sphere chainfluids are presented in section 4.4. The effect of flexibility is studied in binary mixtures oflinear and partially-flexible hard-sphere chain fluids is show in section 4.5. Results for theinfinite dilution solubility of hard spheres in all studied systems are reported in section 4.6.

4.2 Simulation detailsThe isotropic-nematic phase equilibrium of linear and partially-flexible hard-sphere chainfluids was obtained by Monte Carlo simulations in an expanded version of the Gibbs ensem-ble [125]. The method is based on the gradual exchange of molecules by the coordinatedcoupling / decoupling of segments of a fractional molecule between phases as described insection 2.4. Constant volume NVT expanded Gibbs ensemble simulations are used for de-termining the phase equilibria of pure components, while constant pressure NPT expandedGibbs ensemble simulations are used for the calculation of the phase equilibria of binarymixtures.

During a simulation the following trial moves are attempted: displacements, rotations,reptation, configurational-bias partial regrowths (only for partially-flexible molecules)[80],volume changes, identity exchanges [67, 166] (only for mixtures of linear chains), and cou-pling parameter changes. They are selected randomly but with a fixed probability propor-tional to the ratio 100:100:10:100:1:100:1000 respectively. Volume changes are performedisotropically in the logarithm of the volume (for simulations at constant pressure, volumechanges are performed in one subsystem at a time). Simulation boxes for a starting iso-tropic configuration are defined cubic. A rectangular box with edge lengths with a ratio of1:1.1:1.2 are used for initial nematic configurations. Periodic boundary conditions are usedin all simulation boxes. Maximum displacement, rotation, volume, and coupling param-

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4.3 Linear hard-sphere chain molecules

4

55

eter changes are adjusted for a maximum acceptance ratio of 20%. A Monte Carlo cycleis defined by a number of trial moves equal to the number of molecules N in the system,typically in the order of 1×103. The number of Monte Carlo cycles required were typically5×106 for equilibration and 2×106 cycles for production. Isotropic and nematic phases areidentified by the order parameter S 2 defined by Eq 2.3. A value of S 2 close to 0 identifiesthe isotropic phase and a value close to 1 is related to the nematic phase. Configurations aresampled periodically independent of the fractional state. For the finite size systems studiedhere, it is observed that results obtained from end-state sampling do not differ significantlyfrom those obtained from sampling regardless the fractional state. Therefore, all thermo-dynamic properties are calculated based on the number of whole molecules present in thesystem independent of the fractional state.

Simulations start with initial isotropic and nematic configurations. Initial configurationsare obtained from independent constant pressure NPT ensemble simulations that approx-imate the volume of each phase at equilibrium. In the case of pure components, initialestimations of the packing fractions are taken from the previous section on pure linearhard-sphere chain fluids, section 3.3. For the case of mixtures, initial packing fractions,mole fractions, and equilibrium pressures are approximated from the theoretical work ofvan Westen et al. [83]. The initial size of the simulation boxes is dependent on chainlength. The initial edge lengths of both simulation boxes are defined with a value that hasto be larger than the length of the longest molecule in the system. Small systems sizesresult in simulation boxes with at least one edge length close to the length of the molecule,inducing the formation of smectic phases due to restricted positional order perpendicular tothe molecular length. An initial cubic box (isotropic phase) with a size of 25 was sufficientfor all studied systems. For the rectangular box (nematic phase), an initial edge length of25 (the shorter edge) for systems with molecules shorter than 17 beads, and a length of 27for longer molecules was observed to be sufficient.

The solubility of gases is expressed in terms of dimensionless Henry coefficients Eq.2.47. The infinite dilution residual chemical potential is obtained by the Widom test-particleinsertion method as described in section 2.5.2. Equilibrium configurations are sampledevery 1×103 Monte Carlo cycles and a total of 100 test-particle insertions are attempted foreach sample. To identify the effect of connectivity and molecular anisotropy on solubility,relative Henry coefficients are used as defined by Eq. 2.48. The solubility of hard spheres ina hard-sphere chain fluid H0 are calculated from the Boublik-Mansoori-Carnahan-Starling-Leland equation of state [159–161].

4.3 Linear hard-sphere chain moleculesThe isotropic and nematic phase behavior of linear hard-sphere chain fluids of differentlengths was reported in the previous section 3.3. There, the isotropic and nematic packingfractions at equilibrium were approximated from one-phase constant pressure NPT ensem-ble simulations. The isotropic and nematic packing fractions at equilibrium are directlycalculated from two-phase simulations. In Table 4.1 the isotropic and nematic packingfractions at equilibrium for linear hard-sphere 7-mer to 20-mer chain fluids. Packing frac-tion is defined as η = Nmvm/V , where Nm is the total number of segments, vm = πσ3/6the volume of a single segment, and V the volume of the simulation box. It can be noticedthat a maximum in the packing fraction difference exists for the 14-mer. However, due to

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4

56 4 Isotropic-nematic phase equilibrium of hard-sphere chain fluids

Table 4.1: Packing fraction of the isotropic ηI and nematic phase ηN , and packing fraction differences ∆ηN−I

at equilibrium for linear 7-mer to 20-mer chain molecules. Mean values and standard deviations are obtainedfrom at least 10 independent simulation runs.

m ηI ηN ∆ηN−I

7 0.2945 ± 0.0014 0.3033 ± 0.0010 0.0088 ± 0.00178 0.2548 ± 0.0011 0.2650 ± 0.0010 0.0102 ± 0.00159 0.2274 ± 0.0016 0.2391 ± 0.0009 0.0118 ± 0.001810 0.2066 ± 0.0009 0.2228 ± 0.0007 0.0162 ± 0.001111 0.1859 ± 0.0010 0.2000 ± 0.0008 0.0141 ± 0.001312 0.1722 ± 0.0008 0.1903 ± 0.0006 0.0181 ± 0.001013 0.1636 ± 0.0007 0.1859 ± 0.0006 0.0223 ± 0.000914 0.1530 ± 0.0009 0.1766 ± 0.0009 0.0236 ± 0.001315 0.1426 ± 0.0011 0.1657 ± 0.0011 0.0231 ± 0.001616 0.1344 ± 0.0008 0.1574 ± 0.0013 0.0230 ± 0.001517 0.1268 ± 0.0005 0.1497 ± 0.0009 0.0230 ± 0.001018 0.1192 ± 0.0004 0.1401 ± 0.0006 0.0209 ± 0.000719 0.1121 ± 0.0005 0.1311 ± 0.0008 0.0190 ± 0.000920 0.1072 ± 0.0008 0.1229 ± 0.0010 0.0157 ± 0.0013

numerical uncertainties, this maximum can be located between the 13-mer and the 17-mer.In section 3.3, this maximum was identified for a 15-mer and a 16-mer with a value of∆ηN−I = 0.0220. A large difference in the packing fraction between the isotropic and ne-matic phase is relevant for the solubility difference of hard spheres between both phases, asshown below in section 4.6.

4.4 Binary mixtures of linear hard-sphere chain fluidsIn this section, results for the isotropic-nematic phase equilibria of binary mixtures of linearhard-sphere chains are presented. Tables 4.2 to 4.4 show the simulation results for binarymixtures of linear hard-sphere chains. Figs. 4.1, 4.2 and 4.3 show the reduced pressureP∗ and packing fraction η vs. mole fraction of the largest component x2, for mixturesof an 8-mer with a 14-mer, a 16-mer and a 19-mer, respectively. The plus sign is usedto refer to a mixture, e.g. 8-mer+14-mer indicates a binary mixture of an 8-mer with a14-mer. Reduced pressures are defined, relative to the molecular volume of the shortestcomponent (in all cases the 8-mer), by P∗=P v8−mer/kBT , where P is the pressure of thesystem, and v8−mer is the molecular volume of an 8-mer. Simulation results are compared totheoretical predictions obtained from a Vega-Lago rescaled Onsager theory by van Westenet al. [78]. For the larger part of the phase diagrams, excellent agreement between theoryand simulations is obtained, however, for systems very rich in the short component (x2 ≈ 0),a small overestimation of pressure by the theory is observed. The offset is a consequenceof approximations of higher virial coefficients (which are treated by a Vega-Lago rescalingprocedure). For smaller chain lengths, the phase transition is shifted to higher packingfractions. Therefore, any errors introduced by the approximate treatment of the highervirial coefficients become apparent, leading to somewhat larger deviations between theory

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4.4 Binary mixtures of linear hard-sphere chain fluids

4

57

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

P*

x2

(a)

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

η

x2

(b)

Figure 4.1: Isotropic-nematic phase equilibria for the binary mixture 8-mer+14-mer. (a) reduced pressureP∗ vs. mole fraction of the largest component x2, (b) packing fraction η vs. mole fraction of the largestcomponent x2. Black (•) and empty (◦) dots are simulation results for the isotropic and nematic phaserespectively. Broken lines are constant pressure tie-lines. Solid lines are theoretical results obtained from arescaled Onsager theory by van Westen et al. [78].

and simulations [78, 83]. The only available simulation data for the phase equilibria ofmixtures of linear hard-sphere chains is that of Escobedo and de Pablo [94]. In Fig. 4.2, thesimulation results of Escobedo and de Pablo are compared with the present ones and withthe theoretical predictions of van Westen et al. [78]. All results are in good agreement witheach other, validating previous results and the simulation technique presented here.

For all systems studied a phase split into an isotropic and a nematic phase is observed.This phase split is accompanied by a fractionation of the mixture into an isotropic phasericher in the short component and a nematic phase richer in the long component. Phasesplit and fractionation occurs as a consequence of maximizing the total entropy of the sys-tem, balancing orientational, translational, and mixing entropy. In hard systems this entropymaximum is associated to a maximization of the free volume or equivalently to a minimiza-tion of the excluded volume [167, 168]. For a mixture with a specific concentration, at pres-sures below the isotropic-nematic region, orientational and mixing entropy dominate and aone-phase isotropic system is observed. At higher pressures the translational entropy of theisotropic phase is reduced, and a further gain in total entropy is reached by phase split of thesystem into a nematic and an isotropic phase. The loss in orientational entropy due to thephase split and the loss of mixing entropy by the accompanying fractionation are more thancompensated by the gain in translational entropy. Phase split from an isotropic to a nematicphase increases the translational entropy as a consequence of a reduced excluded volumein the nematic phase when chains lay fairly parallel [20, 169, 170]. Fractionation occursdue to a larger tendency to align of the long chains compared to the short chains [171, 172].This tendency is a consequence of a larger excluded volume difference between the isotro-pic and nematic phase for the long chains than for the short chains [158]. This excludedvolume difference increases with chain length, broadening the fractionation of the systeminto a nematic phase richer in long chains and an isotropic phase more depleted of them.

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4

58 4 Isotropic-nematic phase equilibrium of hard-sphere chain fluids

Tabl

e4.

2:Is

otro

pic

(I)an

dne

mat

ic(N

)ph

ase

equi

libria

abi

nary

mix

ture

oflin

ear

8-m

er+1

4-m

erha

rd-s

pher

ech

ain

fluid

sas

afu

nctio

nof

redu

ced

pres

sure

P∗

=P

v 8−

mer/k

BT

.Fo

rea

chph

ase,

mol

efra

ctio

nsof

the

long

chai

nx 2

,pa

ckin

gfra

ctio

nsη

=N

mv m/V

=(N

1∗

m1

+N

2∗

m2)∗

v m/V

,or

der

para

met

ers

S2,

and

redu

ced

pres

sure

sP∗,a

rere

port

ed.

Rep

orte

dst

atis

tical

unce

rtai

ntie

sar

eeq

uiva

lent

toon

est

anda

rdde

viat

ion.

P∗

xI 2S

I 2η

IxN 2

SN 2

ηN

2.65

000.

0000±

0.00

000.

1940±

0.05

520.

2548±

0.00

110.

0000±

0.00

000.

6162±

0.02

670.

2650±

0.00

102.

0000

0.03

70±

0.00

310.

0654±

0.01

110.

2290±

0.00

070.

1573±

0.00

430.

6824±

0.04

520.

2494±

0.00

191.

8000

0.06

69±

0.00

310.

0583±

0.00

770.

2197±

0.00

080.

3593±

0.01

000.

8449±

0.01

350.

2565±

0.00

251.

6000

0.09

51±

0.00

320.

0531±

0.00

650.

2104±

0.00

050.

4060±

0.00

960.

8305±

0.01

160.

2459±

0.00

181.

4000

0.14

70±

0.00

440.

0664±

0.00

540.

2004±

0.00

070.

4764±

0.01

140.

8211±

0.01

080.

2347±

0.00

161.

2000

0.23

36±

0.00

470.

0633±

0.00

390.

1897±

0.00

070.

5671±

0.00

720.

8143±

0.00

870.

2228±

0.00

141.

0000

0.38

37±

0.00

800.

0772±

0.00

860.

1784±

0.00

070.

6815±

0.01

060.

8143±

0.00

930.

2098±

0.00

180.

8000

0.57

80±

0.00

510.

0834±

0.00

710.

1647±

0.00

040.

7681±

0.00

610.

7625±

0.01

220.

1885±

0.00

120.

6000

0.96

60±

0.00

170.

0970±

0.01

240.

1489±

0.00

060.

9812±

0.00

090.

7402±

0.02

080.

1671±

0.00

170.

6229

1.00

00±

0.00

000.

1256±

0.02

890.

1530±

0.00

091.

0000±

0.00

000.

8078±

0.01

130.

1766±

0.00

09

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4.4 Binary mixtures of linear hard-sphere chain fluids

4

59

Table4.3:Isotropic

(I)andnem

atic(N

)phaseequilibria

ofbinarym

ixtureoflinear8-m

er+16-merhard-sphere

chainfluids

asa

functionofreduced

pressureP∗.

Foreach

phase,mole

fractionsofthe

longchain

x2 ,packingfractions

η,orderparameters

S2 ,and

reducedpressures

P∗,are

reported.Reported

statisticaluncertaintiesare

equivalenttoone

standarddeviation.

P∗

xI2

SI2

ηI

xN2

SN2

ηN

2.65000.0000

±0.0000

0.1940±

0.05520.2548

±0.0011

0.0000±

0.00000.6162

±0.0267

0.2650±

0.00102.4000

0.0064±

0.00100.1024

±0.0275

0.2461±

0.00170.0750

±0.0013

0.7734±

0.01410.2691

±0.0012

2.20000.0093

±0.0016

0.0735±

0.01290.2371

±0.0016

0.1802±

0.00330.8395

±0.0059

0.2716±

0.00132.0000

0.0157±

0.00140.0534

±0.0083

0.2284±

0.00090.3799

±0.0108

0.9139±

0.00350.2821

±0.0011

1.80000.0220

±0.0029

0.0484±

0.00530.2191

±0.0010

0.4197±

0.01360.9045

±0.0069

0.2725±

0.00161.6000

0.0398±

0.00250.0485

±0.0046

0.2093±

0.00110.5278

±0.0155

0.9171±

0.00550.2675

±0.0019

1.40000.0685

±0.0046

0.0565±

0.00590.1986

±0.0008

0.5804±

0.01470.9130

±0.0053

0.2551±

0.00161.2000

0.1173±

0.00830.0583

±0.0055

0.1878±

0.00060.6347

±0.0179

0.9023±

0.00500.2412

±0.0017

1.00000.1970

±0.0097

0.0556±

0.00320.1761

±0.0004

0.6782±

0.01740.8859

±0.0080

0.2236±

0.00150.8000

0.3205±

0.01030.0713

±0.0072

0.1625±

0.00030.6930

±0.0170

0.8294±

0.01270.1982

±0.0018

0.60000.6158

±0.0034

0.1362±

0.02390.1485

±0.0007

0.8356±

0.00550.8258

±0.0061

0.1777±

0.00120.4730

1.0000±

0.00000.1962

±0.2124

0.1344±

0.00051.0000

±0.0000

0.7932±

0.13170.1574

±0.0028

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4

60 4 Isotropic-nematic phase equilibrium of hard-sphere chain fluids

Tabl

e4.

4:Is

otro

pic

(I)an

dne

mat

ic(N

)pha

seeq

uilib

riaof

bina

rym

ixtu

reof

linea

r8-m

er+1

9-m

erha

rd-s

pher

ech

ain

fluid

sas

afu

nctio

nof

redu

ced

pres

sure

P∗.

For

each

phas

e,m

ole

fract

ions

ofth

elo

ngch

ain

x 2,p

acki

ngfra

ctio

nsη,o

rder

para

met

ers

S2,

and

redu

ced

pres

sure

sP∗,a

rere

port

ed.R

epor

ted

stat

istic

alun

cert

aint

ies

are

equi

vale

ntto

one

stan

dard

devi

atio

n.

P∗

xI 2S

I 2η

IxN 2

SN 2

ηN

2.65

000.

0000±

0.00

000.

1940±

0.05

520.

2548±

0.00

110.

0000±

0.00

000.

6162±

0.02

670.

2650±

0.00

102.

2000

0.00

23±

0.00

100.

0612±

0.01

160.

2360±

0.00

150.

1240±

0.00

280.

7473±

0.04

030.

2670±

0.00

172.

0000

0.00

49±

0.00

130.

0586±

0.00

620.

2279±

0.00

080.

5240±

0.00

980.

9584±

0.00

170.

3115±

0.00

221.

8000

0.00

67±

0.00

160.

0500±

0.00

300.

2189±

0.00

070.

6226±

0.01

260.

9655±

0.00

250.

3073±

0.00

241.

6000

0.01

40±

0.00

310.

0464±

0.00

390.

2083±

0.00

060.

6967±

0.01

460.

9660±

0.00

250.

2987±

0.00

231.

4000

0.03

17±

0.00

080.

0509±

0.00

360.

1989±

0.00

110.

7637±

0.00

330.

9685±

0.00

130.

2903±

0.00

121.

2000

0.05

97±

0.00

510.

0462±

0.00

150.

1863±

0.00

010.

7936±

0.00

460.

9630±

0.00

090.

2717±

0.00

011.

0000

0.10

51±

0.00

440.

0507±

0.00

380.

1749±

0.00

040.

7870±

0.02

020.

9484±

0.00

650.

2496±

0.00

400.

8000

0.19

45±

0.01

710.

0600±

0.00

410.

1616±

0.00

090.

7904±

0.03

650.

9250±

0.01

320.

2238±

0.00

650.

6000

0.35

52±

0.01

110.

0695±

0.00

530.

1455±

0.00

060.

7855±

0.02

540.

8744±

0.01

690.

1892±

0.00

390.

4000

0.72

32±

0.01

100.

0939±

0.01

160.

1261±

0.00

060.

8809±

0.02

100.

8068±

0.03

470.

1527±

0.00

520.

3500

1.00

00±

0.00

000.

1156±

0.02

430.

1192±

0.00

041.

0000±

0.00

000.

8068±

0.00

720.

1401±

0.00

06

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4.4 Binary mixtures of linear hard-sphere chain fluids

4

61

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

P*

x2

(a)

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

η

x2

(b)

Figure 4.2: Isotropic-nematic phase equilibria for the binary mixture 8-mer+16-mer. (a) reduced pressure P∗

vs. mole fraction of the largest component x2, (b) packing fraction η vs. mole fraction of the largest componentx2. Symbols and lines as in Fig. 4.1. Crosses (×) are simulation results from Escobedo and de Pablo [94].

Increasing fractionation with the difference in molecular length was reported in previousstudies [78, 173, 174] and is also observed in the results shown here.

Figs. 4.1 to 4.3 (a) show a broader isotropic-nematic region, in binary mixtures, as thelength of the long chain (14-mer,16-mer and 19-mer) increases for a constant length of theshort chain (8-mer). The behavior of packing fraction with mole fraction of the largestcomponent is shown in Figs. 4.1 to 4.3 (b). It can be observed that the packing fraction ofthe isotropic phase at equilibrium decreases rapidly with mole fraction. This decrement iscaused by the alignment potential that the long chains introduces in the fluid, facilitating theformation of the nematic phase at lower equilibrium pressures. In the nematic phase, at lowconcentrations of the long chains, the packing fraction remains fairly constant (Fig. 4.1)or increases with mole fraction (Figs. 4.2 and 4.3), although the coexistence pressure de-creases. This behavior is a consequence of the “induced order” [78] introduced by theaddition of a small amount of the long component to a fluid principally composed by shortmolecules. The long chain induce a larger pair excluded volume difference between thenematic and isotropic phase compared to the short chain [158]. This effect leads to a dra-matic increase in the orientational order of the system and an increase in the density of thenematic phase. These two competing effects (reduced pressure vs. induced order) lead tothe observed maximum in the packing fraction of the nematic phase in binary mixtures.

In the theoretical study of van Westen et al. [78], a nematic-nematic region was detectedfor binary mixtures at high reduced pressures. Specifically, it was shown that for the mix-ture 8-mer+19-mer a nematic-nematic region follows the isotropic-nematic equilibria aftera triple point at a reduced pressure of 2.234 [78]. Here, phase equilibria simulations forthe 8-mer+19-mer mixture are performed at high pressures to try to disclose the existenceof the nematic-nematic region. Simulations at a reduced pressure of 2.4 show phase equi-librium between a nematic phase, concentrated in the short chains, and a smectic-A phase,concentrated in the long chains. The smectic phase is formed by a layer of short chain

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4

62 4 Isotropic-nematic phase equilibrium of hard-sphere chain fluids

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

P*

x2

(a)

0.1

0.15

0.2

0.25

0.3

0.35

0 0.2 0.4 0.6 0.8 1

η

x2

(b)

Figure 4.3: Isotropic-nematic phase equilibria for the binary mixture 8-mer+19-mer. (a) reduced pressure P∗

vs. mole fraction of the largest component x2, (b) packing fraction η vs. mole fraction of the largest componentx2. Symbols and lines as in Fig. 4.1.

molecules without clear positional order between two layers of long chain molecules withdefined positional order. At a higher pressure of 3.0, the nematic phase is turned into a phasewhere long chains are locally clustered with a defined orientation and position surroundedby short chains oriented towards the nematic director but with no clear positional order.At this pressure, the smectic-A phase is clearly defined arranging a layer of short chains,formed by two consecutive layers of short chains, between two layers of long chains. Theseresults are considered as preliminary since the systems and corresponding box sizes aretoo small to accommodate smectic phases without any influence of the periodic boundaryconditions on the positional order of the system. Nevertheless, the theoretical existence ofnematic and smectic phases in equilibrium at high pressures has been reported for the caseof hard-spherocylinders [175–177]. And, although, the theoretical results of van Westen etal. do not consider the formation of smectic phases, Cinacchi et al. [178] showed thata metastable nematic-nematic region can precede the formation of stable nematic-smecticequilibria.

4.5 Binary mixtures of linear and partially-flexible hard-sphere chain fluids

Partially-flexible molecules are introduced to study the effect of molecular flexibility on theisotropic-nematic phase equilibria. A partially-flexible molecule is defined as a hard-spherechain molecule with a linear part and a freely-jointed part. The effect of flexibility is inves-tigated in two binary mixtures: a linear 8-mer with a 19-18-mer, and a linear 8-mer with a19-17-mer. Figs. 4.4 and 4.5 show the results for the isotropic-nematic phase equilibria inthese systems. Comparing these results with the fully linear case, Fig. 4.3, it is observedthat flexibility has the effect of, first, increasing the pressure at which the two-phase re-gion starts to appear, and second, reducing the degree of fractionation between coexisting

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4.5 Binary mixtures of linear and partially-flexible hard-sphere chain fluids

4

63

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

P*

x2

(a)

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

η

x2

(b)

Figure 4.4: Isotropic-nematic phase equilibria for the binary mixture of a linear 8-mer with the partially-flexible19-18-mer. (a) reduced pressure P∗ vs. mole fraction of the largest component x2, (b) packing fraction η vs.mole fraction of the largest component x2. Symbols and lines as in Fig. 4.1.

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

P*

x2

(a)

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

η

x2

(b)

Figure 4.5: Isotropic-nematic phase equilibria for the binary mixture of a linear 8-mer with the partially-flexible19-17-mer. (a) reduced pressure P∗ vs. mole fraction of the largest component x2, (b) packing fraction η vs.mole fraction of the largest component x2. Symbols and lines as in Fig. 4.1.

phases. Flexibility decreases the anisotropy of the chain, diminishing the gain in transla-tional entropy that can be obtained from a phase split into a nematic phase. Therefore, acloser packing is needed for the nematic phase to start to form, increasing the pressure atwhich the isotropic-nematic equilibria take place. The reduced fractionation is explainedby a lower excluded volume difference between the isotropic and the nematic phase (lowertendency to align) for the partially-flexible chain compared to the linear case, as shown forthe pair excluded volume of two partially-flexible molecules by van Westen et al. [157].

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4

64 4 Isotropic-nematic phase equilibrium of hard-sphere chain fluids

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

η

Hk

Hk

0

___

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

η

Hk

Hk

0

___

Figure 4.6: Relative infinite dilution solubility Hk/H0k vs. packing fraction η for hard spheres in binary mixtures

of linear and partially-flexible hard-sphere chains in the isotropic and nematic phase at equilibrium. (+) 8-mer+14-mer, (×) 8-mer+16-mer, (�) 8-mer+19-mer, (N) 8-mer+19-18-mer, and (◦) 8-mer+19-17-mer. Thesolid line is a linear regression calculated from all simulation data.

4.6 Solubility of hard spheres in hard-sphere chain fluidsIn the previous chapter, it was showed that a linear relationship exists between relative in-finite dilution solubility, defined by Eq. 2.48, with packing fraction. It was found that thisrelationship is practically independent of chain length for the linear hard-sphere 10-merchain fluid and longer chains fluids. Moreover, it was demonstrated that this relationshipdoes not depend on the liquid crystal state of the fluid, either isotropic or nematic, beingonly a function of packing fraction. In principle, the independence of relative solubility onchain length for long linear chains (longer than 10 segments) can be explained from pairexcluded volume interactions. van Westen et al. [157] derived an expression for the dimen-sionless pair excluded volume between two linear chains of different length V∗ex = Vex/Vm,where Vex is the pair excluded volume and Vm is the volume of a chain of m hard spheres.Here m = (m1 + m2) /2, with m1 and m2 as the lengths of the two linear chains. If one of thechains is considered just as a hard sphere, that expression reduces to V∗ex = (11m − 3) /m. Itcan be observed that as m increases the value of the excluded volume approaches a limitingvalue. For the linear hard-sphere 10-mer chain fluid and longer chain fluids, changes in therelative excluded volume are progressively smaller, explaining in principle the unnoticeableeffect of chain length on solubility reported in the previous chapter. It is remarkable thatthis argument taken from pair molecular interactions also holds for high packing fractionswhere multi-body interactions start to be relevant.

Fig. 4.6 shows the relative infinite dilution solubility of hard spheres in binary mixturesof linear, and linear with partially-flexible hard-sphere chains vs. packing fraction. It canbe observed that, as for the pure component case (Ref. 46), a linear relationship of relativesolubility with packing fraction independent of the mixture type is also obtained. Fig 4.6includes a linear regression for all reported data. The slope and intercept of this regressionare respectively -2.674±0.086 and 1.065±0.015, which are equivalent to the ones reported

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4.6 Solubility of hard spheres in hard-sphere chain fluids

4

65

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1

ln H*

k

x2

Figure 4.7: Dimensionless Henry’s Law constants ln H∗k vs. mole fraction of the longest component x2 forthe binary mixture 8-mer+16-mer in the isotropic and nematic phase at equilibrium. Symbols and lines as inFig. 4.1.

in section 3.5. This result is coherent with the results of solubility in pure components,where for relatively long enough chains solubility seems to be independent of chain length.It has to be noticed that similarly to the pure component case, this linear relationship wouldeventually not be independent on chain length if one of the components forming the mixturebecomes very small.

Finally, Fig. 4.7 shows dimensionless Henry coefficients, defined by Eq. 2.47, vs. molefraction of long chains for the solubility of hard spheres in a mixture of linear chains, 8-mer+16-mer, at the isotropic-nematic coexistence. It can be observed that the solubility ofhard spheres in the isotropic phase increases with mole fraction (indicated by a decreasein Henry’s Law constants), showing a rapid increment at mole fractions close to the pureshort chain fluid. The solubility in the nematic phase is fairly decreased at low values of themole fraction showing a minimum after which it increases monotonically. This behavioris analogous to the changes in packing fraction observed in Fig. 4.2. A maximum in thesolubility difference between the isotropic and the nematic phase is detected at a molefraction of around 0.5, corresponding to the maximum in the packing fraction differencebetween both phases observed at the same concentration.

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4

66 4 Isotropic-nematic phase equilibrium of hard-sphere chain fluids

4.7 ConclusionsThe isotropic-nematic phase equilibria of pure components and binary mixtures of linearand partially-flexible hard-sphere chains were directly calculated from expanded Gibbs en-semble simulations. For pure components, the packing fractions of the isotropic and ne-matic phases at equilibrium were obtained for linear hard-sphere 7-mer to 20-mer chainfluids. These results show a maximum in the packing fraction difference between bothphases for a chain length between the 13-mer and the 17-mer. For binary mixtures, thepacking fraction and mole fraction of the coexisting isotropic and nematic phases wereobtained for mixtures of a linear 8-mer with a 14-, 16- and 19-mer. Phase split, betweenan isotropic and a nematic phase, and fractionation between an isotropic phase richer inthe short component and a nematic phase richer in the large component, is observed forall binary mixtures. The degree of fractionation between both phases increases with thechain length of the largest component for a constant length of the short chain. The effectof molecular flexibility was studied in binary mixtures of an 8-mer with a partially-flexible19-18-mer, and a 19-17-mer. Flexibility increases the isotropic-nematic equilibrium pres-sure and reduces the degree of fractionation between both phases. The relative infinitedilution solubility of hard spheres in the isotropic and nematic phases at equilibrium wasobtained for all studied binary mixtures. A linear relationship between relative solubilityand packing fraction is found. This linear relationship is equivalent to the one obtainedin section 3.5. Binary mixtures show a larger packing fraction difference and, therefore, alarger hard sphere solubility difference between the isotropic and the nematic phase thanthe constituent pure components. This result shows that mixtures of liquid crystals have thepotential of largely increasing the solubility difference of gases between the isotropic andthe nematic phase. A large solubility difference is relevant for the use of liquid crystals assolvents for gas separation applications, as shown in studies that propose liquid crystals asnovel solvents for CO2 capture processes [6, 7].

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5Isotropic-nematic phase equilibrium of Lennard-Jones chainfluids

The liquid crystal phase equilibria of Lennard-Jones chain fluids and the solubility of aLennard-Jones gas in the coexisting phases are calculated from Monte Carlo simulations.Direct phase equilibria calculations are performed using an expanded formulation of theGibbs ensemble. Monomer densities, order parameters, and equilibrium pressures are re-ported for the coexisting isotropic and nematic phases of: (1) linear Lennard-Jones chains,(2) a partially-flexible Lennard-Jones chain, and (3) a binary mixture of linear Lennard-Jones chains. The effect of chain length is determined by calculating the isotropic-nematiccoexistence of linear Lennard-Jones 8-mer, 10-mer, and 12-mer chain fluids. The effect ofmolecular flexibility on the isotropic-nematic equilibrium is studied in a Lennard-Jones 10-mer chain fluid with one freely-jointed segment at the end of the chain. Isotropic-nematicphase split and fractionation are observed for a binary mixture of linear 7-mer and 12-merchains. Simulation results are compared with theoretical predictions as obtained from theanalytical equation of state from the work of van Westen et al. [79]. Excellent agreement be-tween theory and simulations is observed. The solubility of a monomer Lennard-Jones gasin the coexisting isotropic and nematic phases is estimated using the Widom test-particleinsertion method. A linear relationship between solubility difference and density differenceat the isotropic-nematic phase transition is observed. Furthermore, it is shown that gassolubility is independent of the nematic ordering of the fluid, at constant temperature anddensity conditions.

This chapter is based on:B. Oyarzún, T. van Westen, and T.J.H. Vlugt accepted in Molecular Physics [179].

67

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5

68 5 Isotropic-nematic phase equilibrium of Lennard-Jones chain fluids

5.1 IntroductionIn general, a certain degree of molecular anisotropy is required for the appearance of liquidcrystal phases [1, 2]. Early theoretical work, i.e. the work of Onsager [20], Born [180], andMaier and Saupe [21–23], respectively, has shown that both repulsive and attractive inter-actions between molecules can drive the transition to a liquid crystal phase. Nowadays, itis well understood that anisotropic repulsive molecular interactions are a necessary condi-tion for the appearance of ordered phases [29, 31, 32], while both repulsive and attractiveinteractions determine the rich phase behaviour observed in liquid crystal fluids [1, 2].

Simulation and theoretical studies of complex fluids are often based on a coarse-graineddescription of molecules. Coarse-grained models are used to reduce the information ofmolecular structure into main molecular properties, summarized by a small set of modelparameters. Simulations based on coarse-grained models can access longer time and lengthscales than their atomistic counterparts, allowing a bulk description of fluids. Coarse-grained models are commonly used to represent a simplified picture of large moleculessuch as biomolecules [68, 181–184], polymers [110, 185–189], or liquid crystals [47, 48,60, 62, 190, 191]. Moreover, simulation results obtained from coarse-grained models canbe directly compared with theoretical predictions that are based on a well-defined Hamilto-nian, such as the family of perturbation theories developed from the statistical associationfluid theory (SAFT) [69–76]. Traditionally, this relation between fluid theories and molec-ular simulations was primarily used for the development of improved theories. More recentdevelopments show that this is not a one-way street, as accurate SAFT-type theories alsoprovide a very efficient means to derive coarse-grained force fields for use in molecularsimulations [187, 189, 192–197] (see Ref. 188 for a recent review).

Typically, coarse-grained models use simple expressions for the interaction energies.Pair-interaction potentials as hard-sphere [46, 85, 88–90, 154, 164], hard-ellipsoid [35, 37,38], hard-spherocylinders [39, 43, 44], and the Gay-Berne potential [51–55, 198] are pop-ular for studying liquid crystals. Other more elaborated interaction models have been pro-posed, e.g. hard-spherocylinder with an attractive square-well potential [57–60], hard-discwith an anisotropic square-well attractive potential [62], hard-spherocylinder with an attrac-tive Lennard-Jones potential [61], anisotropic soft-core spherocylinder potential [63, 199],and copolymers [64–66].

In this chapter, the isotropic-nematic phase behavior of linear and partially-flexibleLennard-Jones chain fluids is studied. Apart from our recent work [79, 179], there isno other study showing simulation results for the isotropic-nematic phase transition ofLennard-Jones chains. Galindo et al. [95] studied the phase behavior of linear Lennard-Jones chains of 3 and 5 segments; however, in that study, no liquid crystalline phases wereobserved due to the short length of the chains. In this sense, the importance of this chapter istwofold, first to determine the effect of main molecular characteristics (such as chain length,flexibility, attractive interactions and composition of mixtures) on the isotropic-nematic be-havior of long Lennard-Jones chains, and second to validate the analytical equation of statepresented by van Westen et al. [79] for Lennard-Jones chain fluids with variable degree offlexibility.

The equation of state of van Westen et al. was developed using a perturbation the-ory based on a reference fluid of hard-chain molecules. An important assumption in thedevelopment of the equation of state was that of orientation-independent attractive interac-

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5.2 Simulation details and theory

5

69

tions. As shown in Ref. 79, this assumption leads to a surprisingly good description of theisotropic-nematic phase behaviour of a linear Lennard-Jones 10-mer fluid. Here, a moreelaborate evaluation of the equation of state is obtained by comparing it with simulationdata, allowing further analysis on the effect of molecular orientation on attractive disper-sion interactions.

An important property of liquid crystals, relevant for technological applications, is thesolubility of gases in them. Recently, liquid crystals have been proposed as new solventsfor CO2 capture [6–10]. The principle behind this application is described in section 1.2 ofthis thesis. In this chapter, molecular simulations are used to analyze the effect of density,temperature, and composition of mixtures, on the solubility difference of gases between thecoexisting isotropic and nematic phases.

Section 5.2 presents the simulation details for the phase equilibrium calculation ofLennard-Jones chains together with a brief description of the equation of state of van Westenet al.. Simulation results for the isotropic-nematic phase equilibria of linear Lennard-Joneschain fluids are presented in section 5.3. The effect of flexibility on the equilibrium proper-ties is studied in section 5.4 for a partially-flexible Lennard-Jones chain fluid. The isotropic-nematic phase equilibria of a binary mixture of linear Lennard-Jones chains is shown in sec-tion 5.5. Solubility results for a Lennard-Jones gas in the coexisting isotropic and nematicphases are presented in section 5.6. Results are summarized in section 5.7.

5.2 Simulation details and theoryThe isotropic-nematic phase equilibria of linear and partially-flexible Lennard-Jones chainsis studied in this chapter. A chain molecule is defined as a molecule made of spherical seg-ments connected by a rigid segment-to-segment bond length equal to the segment diameterσ. The pair potential between two segments i and j separated by a distance ri j is definedby the Lennard-Jones potential, Eq. 2.2. The depth of the potential well ε and the segmentdiameter σ are constant and equal to 1 for all molecules and systems considered in thiswork. The pressure of the system is calculated by the molecular virial for chain moleculesin a system with periodic boundaries as described by Theodorou et al. [200].

All magnitudes reported here are dimensionless with ε as the basis for energy and σas the basis for length. Some of these magnitudes are: reduced temperature T ∗=kBT/ε,where T is the temperature and kB is the Boltzmann constant; reduced monomer densityρ∗m= mNσ3/V , where N is the number of molecules; and reduced pressure P∗=Pσ3/kBT ,where P is the pressure of the system. In a binary mixture, the reduced monomer densityis defined as ρ∗m= (m1x1 + m2x2)Nσ3/V , where x1 and x2 are the mole fraction of the shortand long chain respectively. Nematic phases are characterized by the order parameter S 2,defined as in Eq. 2.3.

Phase equilibria calculations are performed in an expanded version of the Gibbs en-semble, see section 2.4 and Refs. 120, 121, 164. Constant volume simulations are used forthe calculation of single component systems, while constant pressure simulations are usedin the case of mixtures. At every Monte Carlo step, one of the following trial moves isattempted: displacement, rotation, reptation, configurational-bias partial regrowth (only forpartially-flexible molecules)[80], volume change, identity exchanges[166] (only for mix-tures of linear chains), and coupling parameter change. Trial moves are selected randomlywith a fixed probability proportional to the ratio 100:100:10:100:1:100:1000 respectively.

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5

70 5 Isotropic-nematic phase equilibrium of Lennard-Jones chain fluids

Volume changes are performed isotropically in the logarithm of the volume. Initial isotro-pic configurations are placed in a cubic box, while initial nematic configurations are placedin a rectangular box with an edge length ratio of 1:1.1:1.2. The maximum displacement,rotation, volume, and coupling parameter changes are adjusted for a maximum acceptanceratio of 20%. Periodic boundary conditions are used in all systems. The contribution of thefractional molecules to the tail corrections is neglected as this energy contribution is verysmall. A Monte Carlo cycle consists of a number of trial moves equal to the total numberof molecules in the system (∼1×103). Equilibration requires typically 3×106 cycles andaverages were collected in 2×106 production cycles.

Infinity dilution solubility of gases is expressed as a dimensionless Henry coefficientsdefined by Eq. 2.42. Dimensionless Henry coefficients are related to the infinite dilutionresidual chemical potential of a gas in a solvent [146], which is calculated in constantpressure Monte Carlo simulations using the Widom test-particle insertion method as definedby Eq. 2.57 [148, 152]. The infinite dilution solubility of a single-segment Lennard-Jonesgas molecule is measured in both the coexisting isotropic and nematic phase at equilibrium.Equilibrium configurations are sampled every 1×103 Monte Carlo cycles in which a totalof 100 test-particle insertions are attempted.

The simulation results presented in this chapter are extensively compared to a recentlydeveloped analytical equation of state by van Westen et al. [79]. The equation of state wasdeveloped from a perturbation theory based on a hard-chain reference fluid. The Helmholtzenergy as obtained from the equation of state can thus be written as a sum of differentcontributions according to,

ANkBT

=Aid

NkBT+

Ahc

NkBT+

Adisp

NkBT(5.1)

Here, Aid is an ideal gas contribution, Ahc is a Helmholtz energy due to chain formation,and Adisp is a contribution due to attractive dispersion interactions. The hard-chain term wasobtained from a rescaled Onsager theory, based on the Onsager trial function for describingthe orientational distribution function of the molecules. The dispersion term was developedusing a second order Barker-Henderson theory [201, 202], based on the radial distributionfunction of hard-chain molecules in the isotropic phase. The dispersion contribution istherefore independent on the orientation of the molecules. For details on the equations forcalculating the different contributions, the reader is referred to the work of van Westen etal. [79]. Details on the development of the different contributions to the equation of statecan be found elsewhere [77, 78, 83, 84, 157].

5.3 Linear Lennard-Jones chainsThe liquid crystal phase equilibria of linear Lennard-Jones 8-, 10- and 12-mers were cal-culated from constant volume expanded Gibbs ensemble simulations. Initial estimates ofdensities in the coexisting isotropic and nematic phases were obtained from the equationof state of van Westen et al. [79]. Reduced monomer densities, order parameters, and re-duced pressures at the isotropic-nematic coexistence are shown in Table 5.1 as a function ofreduced temperature. Temperature-density phase diagrams are shown in Fig. 5.1 togetherwith theoretical results obtained from the equation of state. It can be observed that for

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5.3 Linear Lennard-Jones chains

5

71

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

T*

ρ∗

m

(a)

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

T*

ρ∗

m

(b)

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

T*

ρ∗

m

(c)

Figure 5.1: Isotropic-nematic phase equilibria of linear Lennard-Jones chain fluids: (a) 8-mer, (b) 10-mer, and(c) 12-mer. Reduced temperature T ∗ vs. reduced monomer density ρ∗m. Filled (•) and open (◦) symbols aresimulation results for the isotropic and nematic phase respectively. Solid lines are theoretical results obtainedfrom the analytical equation of state of van Westen et al. [79].

longer chains, the isotropic-nematic equilibrium is shifted towards lower densities whilethe density difference between both coexisting phases is increased. Lower equilibrium den-sities and larger density differences are a consequence of larger excluded volume differencesbetween the isotropic and the nematic phase as the chain length increases [158]. For longerchains, the pair-excluded volume is more anisotropic than for shorter chains, resulting in alarger driving force for the isotropic-nematic transition.

As it is shown in Fig. 5.1, the theoretical results as obtained from the perturbation theoryof van Westen et al. provide an overall excellent description of the isotropic-nematic phasediagram. Nevertheless, some minor deviations can be observed. The small but systematicoverestimation of nematic equilibrium densities is expected, since densities in the nematicphase are already overestimated in the description of the hard-chain reference fluid [83].These small overestimations are a result of the approximations made for describing higher(third, fourth, etc.) virial coefficients in the rescaled Onsager approach for hard chain fluids.Equilibrium densities of the isotropic phase are slightly different from simulation results,with an increasing deviation for longer chain lengths. Albeit the deviations are small, theincreasing deviation with chain length cannot be explained from the behavior of the ref-erence fluid [83], since the description of the hard-chain reference system improves forlonger chain lengths. Fig. 5.2 shows the reduced pressure vs. reduced temperature diagramat isotropic-nematic coexistence for linear Lennard-Jones chain fluids. In this figure, it canbe clearly identified that the theoretical description of the isotropic-nematic phase equilibriais less accurate as the chain length increases. Figs. 5.1 and 5.2 suggest that the theoreticaldescription misses a small (positive) contribution to the driving force for the isotropic tonematic phase transition. van Westen et al. assumed that no explicit orientation depen-dent contribution in the dispersive Helmholtz energy was required for a reliable descriptionof the isotropic-nematic phase equilibria of Lennard-Jones chain fluids [79]. The resultsshown here indicate that this assumption may be less justified for longer chains, whereforea stronger effect of anisotropic interactions is expected. Another theoretical assumptionthat could explain the observed deviations is the use of a fixed temperature-independentaspect ratio of the molecules. This assumption is typically needed in a perturbed-chainapproach [84]; however, it reduces the driving force for the isotropic to nematic phase tran-

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5

72 5 Isotropic-nematic phase equilibrium of Lennard-Jones chain fluids

Tabl

e5.

1:Is

otro

pic

(I)an

dne

mat

ic(N

)ph

ase

equi

libria

oflin

ear

Lenn

ard-

Jone

sm

-mer

chai

nflu

ids

asa

func

tion

ofre

duce

dte

mpe

ratu

res

T∗.

For

each

phas

e,re

duce

dm

onom

erde

nsiti

esρ∗ m

,or

der

para

met

ers

S2,

and

redu

ced

pres

sure

sP∗,

are

repo

rted

.R

epor

ted

stat

istic

alun

cert

aint

ies

are

equi

vale

ntto

one

stan

dard

devi

atio

n.

T∗

ρ∗,I m

SI 2

P∗,I

ρ∗,N m

SN 2

P∗,N

m=

810

0.56

0.00

50.

132±

0.05

94.

809±

0.06

20.

614±

0.00

40.

685±

0.05

14.

810±

0.06

59

0.56

0.00

40.

168±

0.05

34.

026±

0.05

80.

607±

0.00

30.

689±

0.05

14.

030±

0.05

98

0.55

0.00

40.

196±

0.06

03.

269±

0.03

10.

602±

0.00

30.

709±

0.01

83.

269±

0.03

07

0.54

0.00

40.

141±

0.04

12.

562±

0.02

50.

604±

0.00

30.

747±

0.02

32.

561±

0.02

66

0.54

0.00

50.

262±

0.07

81.

881±

0.03

30.

600±

0.00

40.

744±

0.03

01.

881±

0.03

35

0.52

0.00

70.

243±

0.08

21.

202±

0.01

80.

614±

0.00

50.

813±

0.02

11.

201±

0.01

8m

=10

100.

446±

0.00

20.

144±

0.04

22.

275±

0.11

40.

523±

0.00

20.

789±

0.00

92.

275±

0.11

29

0.43

0.00

20.

138±

0.04

61.

919±

0.01

70.

526±

0.00

30.

801±

0.01

11.

919±

0.01

58

0.44

0.00

60.

198±

0.08

11.

567±

0.02

50.

537±

0.00

40.

828±

0.00

91.

566±

0.02

47

0.42

0.00

40.

192±

0.07

01.

168±

0.01

60.

542±

0.00

40.

849±

0.00

51.

165±

0.01

56

0.41

0.00

70.

225±

0.05

20.

805±

0.01

20.

573±

0.00

60.

893±

0.00

80.

803±

0.01

3m

=12

120.

376±

0.00

40.

081±

0.02

11.

825±

0.04

40.

463±

0.01

10.

820±

0.02

31.

823±

0.04

511

0.37

0.00

40.

085±

0.01

71.

577±

0.03

60.

467±

0.01

00.

834±

0.01

51.

576±

0.03

610

0.36

0.00

20.

087±

0.01

51.

337±

0.01

20.

472±

0.00

60.

846±

0.01

41.

335±

0.01

39

0.35

0.00

40.

085±

0.02

31.

078±

0.02

30.

479±

0.00

70.

862±

0.00

81.

077±

0.02

38

0.34

0.00

50.

085±

0.01

80.

833±

0.02

60.

493±

0.01

10.

882±

0.01

10.

831±

0.02

67

0.32

0.00

40.

072±

0.00

70.

579±

0.01

40.

520±

0.01

10.

909±

0.01

00.

576±

0.01

5

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5.3 Linear Lennard-Jones chains

5

73

0

1

2

3

4

5

4 6 8 10 12

P*

T*

Figure 5.2: Reduced temperature T ∗ vs. reduced pressure P∗ at the isotropic-nematic coexistence for linearLennard-Jones 8-mer (+), 10-mer (4), and 12-mer (×) chain fluids. Solid lines are theoretical results obtainedfrom the analytical equation of state of van Westen et al. [79].

sition as the aspect ratio should slightly increase with temperature (see e.g. Ref. 199). Anincreased aspect ratio would lead to a more anisotropic molecule and therefore a higherdriving force for the phase transition. In summary, the systematic but small deviations be-tween theory and simulations could be caused by the assumption of orientation-independentattractions or a fixed aspect ratio of the molecules. However, it is clear that this assumptiondo not diminish an accurate description of the phase equilibrium.

Fig. 5.2 shows a linear relationship between pressure and temperature at the isotropic-nematic coexistence for linear Lennard-Jones chain fluids. Further analysis is obtainedconsidering the Clapeyron equation, dP/dT= ∆hI−N/(T∆vI−N), where ∆hI−N is the enthalpychange at the isotropic-nematic phase transition, and ∆vI−N is the molar volume changebetween both phases at coexistence. A constant slope dP∗/dT ∗ indicates a proportionalrelationship between enthalpy change and volume change. It is remarkable that energeticand density effects at the isotropic-nematic phase transition are directly coupled, suggestingthat any energetic change at the phase transition can be described from the isotropic-nematicdensity change and vice versa.

At temperatures lower than the ones reported in Fig. 5.1 and Table 5.1, isotropic-smecticand isotropic-solid phase coexistence is found. Fig. 5.3 shows typical snapshots for a se-quence of nematic, smectic-C, and crystal phases for a linear 10-mer fluid as temperaturedecreases. A similar picture of crystal and smectic-C phases of a 10-mer fluid at a constantmonomer density of ρ∗m = 0.8 was reported by Affouard et al. [203]. In the smectic-C phase,molecules are positioned in layers and oriented with the nematic director tilted with respectto the normal of the layers. Smectic phases were detected by the smectic order parameter(Eq. 2.4) and identified as smectic-C phases by inspecting snapshots of final configurations.Table 5.2 shows the results for the observed coexistence between isotropic and smectic-C phases. These data should be considered only as preliminary due to small differences(∼5%) in the calculated pressure of both coexisting phases. This difference is a conse-

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5

74 5 Isotropic-nematic phase equilibrium of Lennard-Jones chain fluids

(a) (b) (c)

Figure 5.3: Typical snapshots of ordered phases for a linear Lennard-Jones 10-mer chain fluid at coexistencebetween an isotropic phase and: (a) a nematic phase at T ∗ = 6, (b) a smectic-C phase at T ∗ = 5, and (c) acrystal phase at T ∗ = 4. The top row shows parallel views and the bottom row shows perpendicular views tothe largest axis of the simulation box.

quence of the difficulties in arranging the tilted oriented layers in the periodically repeatingrectangular box. In the crystal phases, molecules are arranged in layers with the molec-ular axis pointing in the direction of the normal of the layers. Vega et al. [86] describedthe possible crystal phases of hard-sphere dumbbells from closest packing considerations.These authors identified that dumbbells should be arranged in layers with their axis parallelbut tilted from the normal layer by approximately 35◦. In that study, the stacking of layerswas considered in three different ways, forming an ABAB (hexagonal close packed), or anABC (face-centered cubic) sequence, or a sequence where layers are stacked alternating thetilted angles between successive layers. Galindo et al. [95] considered the ABC sequencein the solid phases of linear Lennard-Jones chains, while Polson et al. [204] suggested thatan AAA (body-centered cubic) stacking with a tilt angle of approximately 33◦ is the sta-ble configuration for the crystal phase of a 6-mer Lennard-Jones fluid with finite bendingpotential. Simulations showing crystal phases are considered only as an indication of theformation of solid phases. This is because of large pressure differences observed betweenboth coexisting phases and the known difficulties in performing direct phase equilibriumcalculations of systems including solids [80, 204–206].

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5.3 Linear Lennard-Jones chains

5

75

Table5.2:

Isotropic(I)and

smectic-C

(Sm

C)phase

equilibriaoflinearLennard-Jones

m-m

erchainfluids.

mT∗

ρ∗,Im

SI2

P∗,I

ρ∗,S

mC

mS

Sm

C2

P∗,S

mC

84

0.468±

0.0210.178

±0.101

0.471±

0.0260.699

±0.031

0.924±

0.0210.447

±0.032

105

0.335±

0.0070.101

±0.009

0.338±

0.0150.688

±0.010

0.965±

0.0050.309

±0.015

126

0.308±

0.0150.087

±0.016

0.372±

0.0300.673

±0.032

0.977±

0.0070.348

±0.024

Table5.3:

Isotropic(I)and

nematic

(N)phase

equilibriaofthe

partially-flexibleLennard-Jones

10-9-merchain

fluid.

T∗

ρ∗,Im

SI2

P∗,I

ρ∗,Nm

SN2

P∗,N

100.568

±0.007

0.227±

0.1204.558

±0.048

0.597±

0.0020.601

±0.035

4.562±

0.0459

0.564±

0.0060.283

±0.123

3.823±

0.0330.599

±0.002

0.685±

0.0213.824

±0.036

80.559

±0.004

0.171±

0.0613.187

±0.029

0.600±

0.0020.710

±0.019

3.189±

0.0237

0.560±

0.0050.265

±0.102

2.518±

0.0210.600

±0.002

0.726±

0.0202.519

±0.022

60.555

±0.010

0.295±

0.0601.831

±0.013

0.601±

0.0020.745

±0.007

1.828±

0.0165

0.543±

0.0060.294

±0.108

1.101±

0.0320.605

±0.002

0.781±

0.0131.105

±0.028

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5

76 5 Isotropic-nematic phase equilibrium of Lennard-Jones chain fluids

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

T*

ρ∗

m

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

T*

ρ∗

m

Figure 5.4: Isotropic-nematic phase equilibria of the partially-flexible Lennard-Jones 10-9-mer compared tothe linear Lennard-Jones 10-mer chain fluid. Reduced temperature T ∗ vs. reduced monomer density ρ∗m.Symbols and solid lines for the partially-flexible 10-9-mer fluid as in Fig. 5.1. Dashed lines are theoreticalresults for the linear 10-mer fluid.

5.4 Partially-flexible Lennard-Jones chainsThe effect of molecular flexibility on the isotropic-nematic phase equilibria of Lennard-Jones chain fluids is studied for a partially-flexible 10-9-mer fluid. Table 5.3 reports thesimulation data and Fig. 5.4 compares the isotropic-nematic coexistence densities of thepartially-flexible 10-9-mer to those of the linear 10-mer fluid. It can be observed that theeffect of flexibility is twofold: (1) it increments the densities at which the phase transitiontakes place, and (2) it reduces the density difference between both phases. These obser-vations can be explained from the fact that the pair-excluded volume of partially flexiblechains is less anisotropic than that of linear chains [157], thereby constituting a smallerdriving force for the isotropic to nematic phase transition.

Another effect of flexibility is the appearance of smectic phases at a temperature lowerthan the one observed for the linear chain fluid. At T ∗=5, isotropic-nematic equilibrium isobserved for the 10-9-mer fluid while isotropic-smectic coexistence was found for the lin-ear 10-mer fluid at the same temperature. Our observations suggest that the temperature ofthe isotropic-nematic-smectic triple point of Lennard-Jones chain fluids decreases with in-creasing flexibility, i.e. decreasing molecular anisotropy. A similar behavior was observedin the case of Gay-Berne fluids with different shape anisotropy [56, 207, 208]. At reducedtemperatures lower than the ones reported in Fig. 5.4 and Table 5.3, smectic and crystalphases were observed, similarly as described in the previous section.

5.5 Binary mixture of linear Lennard-Jones chainsConstant pressure expanded Gibbs-ensemble simulations are performed for determining theisotropic-nematic phase equilibria of a binary mixture of linear Lennard-Jones 7-mer and12-mer chains (7-mer+12-mer). Table 5.4 reports numerical results and Fig. 5.5 shows acomparison between simulation results and theoretical predictions obtained from the ana-

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5.5 Binary mixture of linear Lennard-Jones chains

5

77

Table5.4:Isotropic

(I)andnem

atic(N

)phaseequilibria

ofthebinary

mixture

oflinearLennard-Jones7-m

erand12-m

erchainsatP

∗=

2.535.

x2is

them

olefraction

ofthelong

chain(12-m

er).

T∗

xI2

ρ∗,Im

SI2

xN2

ρ∗,Nm

SN2

140.971

±0.001

0.402±

0.0120.143

±0.081

0.988±

0.0010.489

±0.000

0.844±

0.00213

0.766±

0.0110.407

±0.003

0.157±

0.0520.904

±0.007

0.505±

0.0010.837

±0.010

120.606

±0.011

0.418±

0.0020.098

±0.009

0.845±

0.0080.530

±0.001

0.857±

0.00611

0.463±

0.0180.431

±0.003

0.092±

0.0200.787

±0.013

0.556±

0.0020.874

±0.008

100.338

±0.015

0.449±

0.0030.065

±0.010

0.767±

0.0230.603

±0.003

0.907±

0.0109

0.222±

0.0170.467

±0.003

0.065±

0.0060.716

±0.040

0.637±

0.0040.920

±0.014

80.124

±0.010

0.491±

0.0020.062

±0.008

0.668±

0.0410.680

±0.002

0.938±

0.0117

0.050±

0.0010.522

±0.001

0.063±

0.0080.521

±0.009

0.706±

0.0010.938

±0.004

60.014

±0.004

0.566±

0.0010.092

±0.023

0.402±

0.0410.788

±0.048

0.959±

0.0135.397

0.000±

0.0000.606

±0.004

0.156±

0.0400.000

±0.000

0.631±

0.0040.517

±0.081

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5

78 5 Isotropic-nematic phase equilibrium of Lennard-Jones chain fluids

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1

T*

x2

(a)

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

ρ∗

m

x2

Figure 5.5: Isotropic-nematic phase equilibria for a binary mixture of linear Lennard-Jones 7-mer and 12-merchains at a reduced pressure of P∗ = 2.535. (a) Reduced temperature T ∗ and (b) reduced monomer densityρ∗m vs. mole fraction of the largest component x2. Symbols and solid lines as in Fig. 5.1. Dashed linesare constant temperature tie-lines as reported in Table 5.4. Pure component simulation data for the 7-merfluid are calculated from constant volume expanded Gibbs-ensemble simulations at temperatures close to theequilibrium temperature. Equilibrium temperature and isotropic-nematic coexisting densities at the specifiedpressure of the mixture are determined by interpolation from the closest calculated equilibrium pressures.

lytical equation of state of van Westen et al. [79]. Phase split between an isotropic and anematic phase and fractionation of the fluid between both phases is observed. Fractionationof the mixture into an isotropic phase richer in the short chains and a nematic phase richerin the long chains is a consequence of the more anisotropic pair-excluded volume interac-tions of the long chains. Fig. 5.5 (a) shows the mole fraction of the long chain x2 in theisotropic and nematic phases at different equilibrium temperatures. An accurate descriptionof fractionation between both phases is obtained from theoretical calculations. Fig. 5.5 (b)compares simulation results for the isotropic and nematic coexistence monomer densitieswith theoretical results from the equation of state. At constant mole fraction, a maximumin the density of the nematic phase and a maximum in the isotropic-nematic density differ-ence is observed in the range x2=0.4−0.6 from simulation results and at x2=0.5 from thetheoretical results. Previously, a similar behavior was found for binary mixtures of linearhard-sphere chains [78, 164]. With increasing mole fraction of the long component, thedriving force for the phase transition increases, which leads to lower coexistence densi-ties. However, the addition of a small amount of the long component to a pure fluid ofthe short component leads to a dramatic increase in the orientational order of the system(due to induced order [78]). As a result, the density of the nematic phase increases. Thesetwo competing effects lead to the observed maximum. Overall, comparison between theoryand simulations is accurate. A slight overestimation of the isotropic and nematic monomerdensities is observed, with larger deviations at lower temperatures corresponding to a largerconcentration of the short chains. This effect is expected since the description of the hard-chain reference fluid that underlies the perturbation theory becomes less accurate for shorterchain lengths [78].

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5.6 Solubility of gases in Lennard-Jones chains

5

79

0.4

0.6

0.8

1

1.2

1.4

0.3 0.4 0.5 0.6

lnH*

k

ρ∗

m

Figure 5.6: Infinite dilution solubility of a Lennard-Jones gas in the coexisting isotropic and nematic phases ofa linear Lennard-Jones 12-mer chain fluid. Symbols and solid lines as in Fig. 5.1. Dashed lines are constanttemperature tie-lines as reported in Table 5.1.

5.6 Solubility of gases in Lennard-Jones chainsIn this section, the infinite dilution solubility of a Lennard-Jones gas in the coexisting iso-tropic and nematic phases of Lennard-Jones chain fluids is studied. Solubility is describedby the dimensionless Henry coefficient as defined by Eq. 2.47.

In Fig. 5.6, simulation results for the solubility of a Lennard-Jones gas in the coexistingisotropic and nematic phases of a linear Lennard-Jones 12-mer chain fluid are reported asa function of monomer density. A decrease in solubility with density is observed in theisotropic phase, while first an increasing and then a decreasing solubility is identified inthe nematic phase. As temperature increases (see Fig. 5.1 (c)), decreasing solubilities inthe isotropic phase are caused by both larger equilibrium temperatures and higher equilib-rium densities, while the increasing/decreasing behavior in the nematic phase is the conse-quence of a balance between larger temperatures accompanied by lower equilibrium den-sities. Larger solubility differences between both coexisting phases are observed at lowerequilibrium temperatures.

Fig. 5.7 shows the infinite dilution solubility of a Lennard-Jones gas in the coexistingisotropic and nematic phases of a binary mixture of linear 7- and 12-mer at constant pres-sure. It can be identified that the solubility difference between both phases is always largerin the mixture than in the pure components. A maximum in the isotropic-nematic solubilitydifference is observed at a mole fraction in the range x2 = 0.4 − 0.6 from simulations andx2 = 0.5 from theoretical results. This maximum is directly related to the maximum densitydifference shown in the previous section, see Fig. 5.5 (b).

Fig. 5.8 shows the isotropic-nematic solubility difference as a function of the densitydifference at coexistence for the linear Lennard-Jones chain fluids. A linear relationshipbetween solubility difference and density difference is identified in all cases. This result in-dicates the central role that density has in solubility, which seems to be independent of the

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5

80 5 Isotropic-nematic phase equilibrium of Lennard-Jones chain fluids

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

lnH*

k

x2

Figure 5.7: Infinite dilution solubility of a Lennard-Jones gas in the coexisting isotropic and nematic phasesof a binary mixture of linear Lennard-Jones 7-mer and 12-mer chains. Logarithm of dimensionless Henry’scoefficient ln H∗k vs mole fraction of the long chain x2. Symbols and solid lines as in Fig. 5.1. Dashed linesare constant temperature tie-lines as reported in Table 5.4.

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25 0.3

∆lnH*

k

∆ρ∗m

Figure 5.8: Solubility difference ∆ ln H∗k as a function of reduced monomer density difference ∆ρ∗m at theisotropic-nematic phase transition for linear Lennard-Jones 8-mer (+), 10-mer (4) and 12-mer (×) chain fluids.Solid lines are theoretical results obtained from the analytical equation of state of van Westen et al. [79].

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5.7 Conclusions

5

81

0.96

0.98

1

1.02

1.04

1.06

0 0.2 0.4 0.6 0.8 1

lnH*

k

S2

Figure 5.9: Infinite dilution solubility of a Lennard-Jones gas in a linear 12-mer fluid at the isotropic-nematicphase transition as a function of nematic ordering. Logarithm of dimensionless Henry’s coefficient ln H∗k vs.order parameter S 2. Results are obtained at constant temperature T ∗ = 10 and monomer density ρ∗m = 0.42.Error bars represent one standard deviation.

nematic ordering of the fluid. To confirm this supposition, simulations at constant numberof molecules, volume, and temperature, in the NVT ensemble, but restricting the nematicordering of the system are performed, similarly as in section 3.5. Fig. 5.9 shows the solubil-ity of gases in a 12-mer for different values of the order parameter at constant temperatureand density. From these results, it can be concluded that the solubility of a Lennard-Jonesgas molecule is independent of the nematic ordering of the fluid at constant density andtemperature conditions.

Figs. 5.6 and 5.7 include theoretical results obtained from the equation of state ofvan Westen et al. for the same temperature range considered in the simulations (see Ta-bles 5.1 and 5.4). These results indicate that the theory systematically underestimates theHenry coefficient. In Fig. 5.8, we show the isotropic-nematic solubility difference as a func-tion of density difference. It can be observed, that theory and simulations follow the samelinear relationship. This underlines the fact that the differences between simulations andtheory are systematic. Unfortunately, we have not found the underlying reason for thesesystematic differences. In Fig. 5.8, it can be observed that density differences are overes-timated by the theory, with the consequence of a higher solubility difference between theisotropic and nematic phase. The overestimation of the density difference at the isotropic-nematic transition is a common flaw of rescaled Onsager theories of the type examined inthe work of van Westen et al., and is expected to become less pronounced for longer chainlengths.

5.7 ConclusionsThe isotropic-nematic phase equilibria of Lennard-Jones chain fluids were determined fromdirect phase equilibria calculations using an expanded version of the Gibbs ensemble. Re-sults for linear Lennard-Jones chain fluids showed that increasing chain length leads to a

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82 5 Isotropic-nematic phase equilibrium of Lennard-Jones chain fluids

decrease of the isotropic-nematic coexistence densities and an increase of the isotropic-nematic density difference. These results are explained from a more anisotropic pair-excluded volume of longer chains, which constitutes a larger driving force for the isotropicto nematic transition. A linear relationship between coexistence pressure and temperaturewas found, indicating a proportional relationship between energetic and volume effects atthe isotropic-nematic phase transition. We found evidence of the formation of smectic-C and crystal phases. However, these results are not conclusive due to the difficulties ofthe current simulation technique in representing the bulk behaviour of highly structuredphases. The effect of flexibility was studied by means of partially flexible chains, con-sisting of a rigid linear part, and a freely-jointed part. It was found that flexibility shiftsthe isotropic-nematic equilibrium to larger densities and pressures, reducing at the sametime the isotropic-nematic density difference. This can be explained from the fact that flex-ibility reduces the anisotropy of molecules, and thereby lowers the driving force for theisotropic-nematic transition. Results for the isotropic-nematic coexistence of a binary mix-ture showed fractionation of the fluid into an isotropic phase richer in the short chains anda nematic phase richer in the long chains. For this mixture, a maximum in the density ofthe nematic phase and a maximum in the isotropic-nematic density difference at coexis-tence were identified. This maximum appears as a balance between a larger potential forthe phase transition as the mole fraction of the long component increases (reducing the co-existing isotropic density), and the order induced in the nematic phase by a larger presenceof the long component (increasing the coexisting nematic density).

Simulation results were extensively compared to theoretical results as obtained froma recently developed equation of state of van Westen et al. [79]. It was found that theequation of state provides an excellent description of phase equilibria. The equation ofstate was developed based on the assumption that attractive dispersive interactions betweenLennard-Jones chains do not depend on their relative orientations. The good comparison tosimulation results therefore suggests that the effect of anisotropic dispersion interactions issmall, and that a faithful description of real nematic liquid crystals could well be developedwithout it.

The infinite dilution solubility of a Lennard-Jones gas in Lennard-Jones chain fluidswas estimated by the Widom test-particle insertion method. Pure components showed anincreasing solubility difference between coexisting isotropic and nematic phases as temper-ature decreases. A linear relationship between solubility difference and density differenceat coexistence was identified for all linear Lennard-Jones systems. In the binary mixture,a maximum in the isotropic-nematic solubility difference as a function of mole fractionwas observed. This maximum is directly related to the observed maximum in the densitydifference between both phases. Simulations at constant temperature and density but withrestricted values of the order parameter showed that the solubility of a Lennard-Jones gasis independent of the nematic ordering of the fluid.

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[193] C. Avendaño, T. Lafitte, A. Galindo, C. S. Adjiman, G. Jackson, and E. A. Müller, SAFT-γ Force Fieldfor the Simulation of Molecular Fluids. 1. A Single-Site Coarse Grained Model of Carbon Dioxide, TheJournal of Physical Chemistry B 115, 11154 (2011).

[194] T. Lafitte, C. Avendaño, V. Papaioannou, A. Galindo, C. S. Adjiman, G. Jackson, and E. A. Müller,SAFT-γ Force Field for the Simulation of Molecular Fluids: 3. Coarse-Grained Models of Benzene andHetero-Group Models of n-Decylbenzene, Molecular Physics 110, 1189 (2012).

[195] O. Lobanova, C. Avendaño, T. Lafitte, E. A. Müller, and G. Jackson, SAFT-γ Force Field for theSimulation of Molecular Fluids: 4. A Single-Site Coarse-Grained model of Water Applicable over a WideTemperature Range, Molecular Physics 113, 1228 (2015).

[196] A. Hemmen, A. Z. Panagiotopoulos, and J. Gross, Grand Canonical Monte Carlo Simulations Guided byan Analytic Equation of State—Transferable Anisotropic Mie Potentials for Ethers, The Journal ofPhysical Chemistry B 119, 7087 (2015), http://dx.doi.org/10.1021/acs.jpcb.5b01806 .

[197] A. Hemmen and J. Gross, Transferable Anisotropic United-Atom Force Field Based on the Mie Potentialfor Phase Equilibrium Calculations: n-Alkanes and n-Olefins, The Journal of Physical Chemistry B 119,11695 (2015).

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[199] A. Cuetos and B. Martínez-Haya, Liquid Crystal Phase Diagram of Soft Repulsive Rods and its Mappingon the Hard Repulsive Reference Fluid, Molecular Physics 113, 1137 (2015).

[200] D. N. Theodorou, T. D. Boone, L. R. Dodd, and K. F. Mansfield, Stress Tensor in Model PolymerSystems with Periodic Boundaries, Macromolecular Theory and Simulations 2, 191 (1993).

[201] J. A. Barker and D. Henderson, Perturbation Theory and Equation of State for Fluids: The Square-WellPotential, The Journal of Chemical Physics 47, 2856 (1967).

[202] J. A. Barker and D. Henderson, Perturbation Theory and Equation of State for Fluids. II. A SuccessfulTheory of Liquids, The Journal of Chemical Physics 47, 4714 (1967).

[203] F. Affouard, M. Kröger, and S. Hess, Molecular Dynamics of Model Liquid Crystals Composed ofSemiflexible Molecules, Physical Review E 54, 5178 (1996).

[204] J. M. Polson and D. Frenkel, Calculation of Solid-Fluid Phase Equilibria for Systems of Chain Molecules,The Journal of Chemical Physics 109, 318 (1998).

[205] R. Agrawal and D. A. Kofke, Thermodynamic and Structural Properties of Model Systems at Solid-FluidCoexistence, Molecular Physics 85, 23 (1995).

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Summary

The liquid crystal phase behavior of chain fluids and the solubility of gases in them wasdetermined using Monte Carlo simulations. The effect of molecular properties such aselongation and flexibility was studied using linear and partially-flexible chain molecules.Linear chain molecules are rigid molecules with all segments aligned on the same molec-ular axis, while partially-flexible molecules are made of a linear and a freely-jointed partthat provide a certain degree of flexibility to the molecule. Volumetric and thermal effectsare studied independently by hard-sphere and Lennard-Jones potentials. Molecular sim-ulations were performed for hard-sphere and Lennard-Jones chain fluids with a focus onthe isotropic-nematic phase transition. The solubility of gases in liquid crystal phases andspecially the solubility change at the isotropic-nematic transition is of interest for gas sep-aration applications. This was studied using a single-segment gas molecule at the infinitedilution limit.

Monte Carlo simulations were performed in the NPT ensemble and in an expanded ver-sion of the Gibbs ensemble. The phase behavior of single-phase hard-sphere chain fluidswas determined using NPT ensemble simulations, while the isotropic-nematic phase equi-librium of single-component and binary mixtures of hard-sphere and Lennard-Jones fluidswas calculated using expanded Gibbs ensemble simulations. Fluid properties such as pack-ing fractions (hard-sphere fluids), densities and temperatures (Lennard-Jones fluids), orderparameters, pressures, and gas solubilities (Henry coefficients) were obtained for chain flu-ids with different degrees of elongation and flexibility.

The study of single-component athermal (hard-sphere) systems showed that larger chainlengths increase the potential for the formation of nematic phases, decreasing monoton-ically the pressure and packing fractions at which the isotropic-nematic phase transitiontakes place. However, a maximum in the isotropic-nematic packing fraction difference wasobserved as a function of chain length. Flexibility reduces the anisotropy of the molecule,which increases the isotropic-nematic transition pressure and decreases the packing frac-tion difference between both phases. It was shown that the solubility of a single-segmenthard-sphere gas molecule does not depend on molecular order, being determined only bythe packing fraction of the system. The solubility difference at the isotropic-nematic phasetransition is therefore determined by the packing fraction difference at coexistence. Theeffect of connectivity was studied by considering relative Henry coefficients, defined as theHenry coefficient of the gas in the chain fluid divided by the Henry coefficient of the gas inthe monomer fluid at same density. A linear relationship between relative Henry coefficientsand packing fraction was obtained.

Binary mixtures of hard-sphere chain fluids showed a phase split and fractionation be-tween a nematic phase concentrated in the longest component and an isotropic phase richerin the shortest component. Fractionation is a consequence of the larger tendency to alignof the long chains compared to the short chains. This tendency increases with the differ-ence in chain length between both components forming the mixture. A larger fractionationleads to a larger isotropic-nematic packing fraction difference, which was found to be al-ways larger for the mixture than for the constituting pure components. Flexibility of the

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94 Summary

longest component reduces fractionation, increasing the pressure and packing fraction atwhich the isotropic-nematic transition takes place for a specific composition of the fluid.As in the case of pure components, a linear relationship between relative Henry coefficientsand packing fraction independent of the molecular order was observed.

The isotropic-nematic phase equilibria of Lennard-Jones chains was determined for:linear chain molecules with different lengths, a partially-flexible chain molecule, and abinary mixture of linear chains. Increasing chain lengths displaces the isotropic-nematiccoexistence of linear Lennard-Jones chain fluids to larger equilibrium densities and lowerequilibrium pressures, increasing at the same time the isotropic-nematic density difference.Flexibility reduces the anisotropy of the molecule shifting the isotropic-nematic equilib-rium to larger pressures and densities, reducing at the same time the density differencebetween both phases. In a binary mixture of linear Lennard-Jones chains, fractionationbetween an isotropic phase richer in the short chains and a nematic phase richer in thelong chains was observed. The isotropic-nematic density difference was found to be al-ways larger in the mixture than for pure components, with a maximum at a certain molefraction. The solubility of a single-segment Lennard-Jones gas molecule was calculatedfor linear Lennard-Jones chain fluids and for a binary mixture of them. A linear relation-ship between solubility difference and density difference at coexistence was identified forall linear chain systems. Furthermore, it was shown that the solubility of a Lennard-Jonesgas in linear Lennard-Jones chain fluids is independent of the molecular order of the fluid atcoexistence. The isotropic-nematic solubility difference in the binary mixture was observedto be always larger in the mixture than for the pure components, with an observed maxi-mum at the mole fraction corresponding to the maximum in the isotropic-nematic densitydifference.

Simulation results for the isotropic-nematic transition were compared to theoretical pre-dictions obtained from the analytical equation of state developed by van Westen et al.,(Refs. 77–79, 83, 84). Excellent agreement between simulation and theoretical results wereobserved. A rescaled Onsager theory was used in the description of hard-sphere systemsshowing an accurate prediction compared to simulation results. A perturbed theory ap-proach was used in the treatment of Lennard-Jones fluids, whereby orientation dependentattractions were not considered explicitly in the development of the dispersion term. A re-liable description of the isotropic-nematic phase equilibrium of Lennard-Jones fluids wasobtained using this approach.

The simulation data presented in this work represents a large contribution to the liquidcrystal phase behavior of chain fluids. The effect of molecular properties on the phasebehavior of liquid crystals was determined using linear and partially-flexible molecules withdifferent elongations and flexibility, for purely repulsive (hard-sphere) and soft-attractive(Lennard-Jones) interactions. Predictions obtained from the equation of state of van Westenet al. showed an excellent agreement with simulation data, validating the assumptions madein the development of this theory.

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Samenvatting

Het vloeibaar-kristallijne fasegedrag van ‘fluids’ bestaande uit ketenmoleculen, alsmede deoplosbaarheid van gassen in deze ‘fluids’, is verkregen uit Monte Carlo simulaties. Het ef-fect van moleculaire eigenschappen zoals elongatie en flexibiliteit is bestudeerd door mid-del van lineaire en deels flexibele ketenmoleculen. Lineaire ketenmoleculen zijn rigide,met alle segmenten uitgelijnd op dezelfde moleculaire as. Deels flexibele ketenmoleculenzijn opgebouwd uit een lineair- en volledig flexibel deel. Het laatste deel verschaft eenzekere maat van flexibiliteit aan het molecuul. Volumetrische en thermische effecten zijnonafhankelijk van elkaar bestudeerd door middel van respectievelijk, een ‘harde-bollen’ en‘Lennard-Jones’ potentiaal. Monte Carlo simulaties zijn uitgevoerd voor zowel ketens vanharde segmenten als segmenten met Lennard-Jones interacties; de focus hierbij was op deisotroop-nematische faseovergang. De oplosbaarheid van gassen in vloeibare kristallen -vooral het oplosbaarheidsverschil tussen twee co-existerende isotrope en nematische fasen- is van belang voor gasabsorptie. Deze oplosbaarheid is bestudeerd voor een oneindigverdunde oplossing van gasmoleculen bestaande uit een enkel segment.

Monte Carlo simulaties zijn uitgevoerd in het NPT ensemble en in een ‘expanded’ ver-sie van het Gibbs ensemble. De NPT simulaties zijn gebruikt om het één-fasegedrag vanharde ketens te bepalen. De simulaties in het Gibbs ensemble zijn gebruikt om de isotroop-nematische faseovergang van zowel pure componenten als mengsels van harde ketens enLennard-Jones ketens te bepalen. Vloeistofeigenschappen zoals pakkingsfracties (hardeketens), dichtheden en temperatuur (Lennard-Jones ketens), orde parameters, druk en op-losbaarheden van gassen (Henry coëfficiënten), zijn verkregen voor ketenfluids van varië-rende elongatie en flexibiliteit.

De studie van athermische (hardebollenpotentiaal) pure systemen, laat zien dat een toe-nemende ketenlengte resulteert in een grotere drijvende kracht voor de isotroop-nematisch-overgang, wat leidt tot een lagere druk en pakkingsfractie bij de faseovergang. Echter,het dichtheidsverschil tussen beide fasen toont een maximum als functie van ketenlengte.Flexibiliteit verkleint de anisotroopheid van een molecuul. Dit leidt tot een hogere over-gangsdruk en een kleiner dichtheidsverschil tussen beide fasen. We tonen aan dat de op-losbaarheid van een gasmolecuul bestaande uit een enkel segment volledig wordt bepaalddoor pakkingsfractie van het systeem en dus niet afhangt van moleculaire ordening. Het op-losbaarheidsverschil bij de isotroop-nematisch-overgang wordt daarom uitsluitend bepaalddoor het dichtheidsverschil tussen beide fasen. Het effect van ketenconnectiviteit is bepaalddoor relatieve Henrycoëfficiënten te beschouwen, gedefinieerd als de Henrycoëfficiënt vanhet gas in de ketenfluid gedeeld door de Henrycoëfficiënt van het gas in een fluid van lossesegmenten bij dezelfde dichtheid. We vinden een lineaire verband tussen relatieve Henrycoëfficiënten en de pakkingsfractie.

Binaire systemen van fluids bestaande uit harde ketenmoleculen tonen een faseschei-ding en fractionering tussen een nematische fase geconcentreerd in de lange component eneen isotrope fase geconcentreerd in de korte component. Fractionering is een gevolg van hetfeit dat langere moleculen een grotere neiging hebben tot ordenen van de moleculaire as.Deze neiging wordt groter naarmate het verschil in ketenlengte groter wordt. Een grotere

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96 Samenvatting

fractionering leidt tot een groter dichtheidsverschil bij de isotroop-nematisch-overgang. Ditdichtheidsverschil is altijd groter dan dat voor de pure componenten. Flexibiliteit van delangste component verkleint de fractionering, wat leidt tot een hogere druk en dichtheid bijde faseovergang. Voor pure systemen vinden we een linear verband tussen relatieve Henry-coëfficiënten en pakkingsfractie. Deze relatie is onafhankelijk van moleculaire ordening.

Het evenwicht tussen isotrope en nematische fasen van Lennard-Jones ketenmoleculenis uitgerekend voor de volgende systemen: lineaire (rigide) ketens van verschillende lengte,een deels flexibel ketenmolecuul, en een binair mengsel van lineaire ketens. Een grotereketenlengte leidt tot (1) een verschuiving van de isotroop-nematisch-overgang naar lageredichtheid en druk, en (2) een kleiner dichtheidsverschil tussen beide fasen. Een grotereflexibiliteit resulteert in een kleinere anisotroopheid van de moleculen. Dit leidt tot (1)een verschuiving van de isotroop-nematisch-overgang naar hogere druk en dichtheid, en(2) een afname van het dichheidsverschil tussen de isotrope en nematische fase. In eenbinair mengsel van lineaire Lennard-Jones ketens is een fractionering van de componententussen beide fasen vastgesteld. Het dichtheidsverschil tussen de isotrope en nematische fa-sen is altijd groter dan dat voor de pure componenten. Het oplosbaarheidsverschil van eenLennard-Jones gasmolecuul is berekend voor een oplosmiddel van lineaire Lennard-Jonesketenmoleculen en voor een binair mengsel van deze moleculen. Voor alle bestudeerde sys-temen is een lineaire relatie tussen het oplosbaarheidsverschil en het dichtheidsverschil bijde faseovergang verkregen. Het is aangetoond dat de oplosbaarheid onafhankelijk is van demoleculaire ordening bij de faseovergang. Het isotroop-nematisch oplosbaarheidsverschilin het binaire systeem is altijd groter dan dat voor de pure componenten.

Simulatieresultaten voor de isotroop-nematisch-overgang zijn vergeleken met theore-tische voorspellingen verkregen uit de analytische toestandsvergelijking van van Westenet al., (Refs. 77–79, 83, 84). We vinden een nauwkeurige overeenstemming tussen simu-laties en theorie. De theorie is gebaseerd op een geschaalde Onsager theorie voor hardeketens, uitgebreid met een perturbatieterm voor het beschrijven van Lennard-Jones ketens.Orientatie-afhankelijke attractieve interacties tussen de moleculen zijn niet meegenomen inde perturbatieterm. Zowel voor harde- als Lennard-Jones ketenmoleculen leidt deze aanpaktot een goede beschrijving van het isotroop-nematische fasegedrag.

De simulatiedata zoals gepresenteerd in dit werk levert een grote bijdrage aan het begripvan het vloeibaar-kristallijne fasegedrag van ketenmoleculen. Het effect van moleculaire ei-genschappen op het fasegedrag van vloeibare kristallen is bepaald door gebruik te makenvan lineaire en deels flexibele moleculen van variërende lengte en flexibiliteit. Zowel ketensmet repulsieve- (gemaakt uit harde bollen) en zacht-attractieve (Lennard-Jones) interactiestussen de segmenten zijn beschouwd. Voorspellingen verkregen uit de toestandsvergelij-king van van Westen et al. komen uitstekend overeen met de resultaten uit de simulaties.Deze goede overeenstemming valideert de benaderingen die gemaakt zijn in de ontwikke-ling van de toestandsvergelijking.

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Curriculum Vitæ

Bernardo Andrés Oyarzún Rivera16-12-1979, born in Viña del Mar, Chile

EDUCATIONDates: October 2007 - October 2009Title: Master in Process and Energy Engineering (Dipl.-Ing.)Thesis: Simulation and economic evaluation of cryogenic CO2 captureInstitution: Technische Universität BerlinCity/Country: Berlin, Germany

Dates: March 1999 - September 2007Title: Bachelor and Master in Chemical Engineering (Ing.-Civ.)Thesis: Lycopene extraction with supercritical CO2Institution: Universidad Técnica Federico Santa MaríaCity/Country: Valparaíso, Chile

Dates: October 1995 - July 1997Title: Secondary EducationInstitution: Colegio Alemán QuitoCity/Country: Quito, Ecuador

PROFESSIONAL EXPERIENCE

InternshipsDates: July 2006 - March 2007Project: Modelling of CO2 scrubbing by enhanced potash solutionsEmployer: BASF S.E.City/Country: Ludwigshafen, Germany

Dates: January 2004 - February 2004Project: Estimation of critical production parameters in the salmon industryEmployer: Salmones Multiexport Ltda.City/Country: Puerto Montt, Chile

Dates: January 2003 - February 2003Project: Technical-economical study of a wastewater treatment plantEmployer: Oxiquim S.A.City/Country: Viña del Mar, Chile

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98 Curriculum Vitæ

Research AssistantDates: November 2009 - January 2011Project: Integrated solvent and process designEmployer: Delft University of TechnologyCity/Country: Delft, The Netherlands

Dates: February 2008 - March 2009Project: Design and development of experimental assets for basic studies on the

coalescence of water/oil/water emulsionsEmployer: Technische Universität Berlin - Sulzer Chemtech Ltd.City/Country: Berlin, Germany

Dates: October 2005 - July 2006Project: Simulation of thermal separation processesEmployer: Technische Universität BerlinCity/Country: Berlin, Germany

OTHER

Workshops

• Advanced Course on Thermodynamic Models - Fundamentals & Computational Aspects,by Michel Michelsen, Technical University of Denmark, Lyngby, Denmark, January2014.

• MolSim 2011: Understanding Molecular Simulation, by Daan Frenkel (Cambridge Uni-versity) and Berend Smit (Berkeley University), University of Amsterdam, Amsterdam,The Netherlands, January 2011.

Prizes and Awards

• DPTI 2nd Annual Event, Poster award: “The effect of molecular shape on the infinitedilution solubility of gases in liquid crystals", Rotterdam, The Netherlands, November2014.

• DAAD (German Academic Exchange Service), fellow for engineering studies in Ger-many, Berlin, Germany, July 2005 - June 2006.

• Universidad Técnica Federico Santa María, member of the university’s honour’s list foroutstanding academic performance, Valparaíso, Chile, 2002 - 2007.

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List of Journal Publications

6. Oyarzún, B.; van Westen, T.; Vlugt, T. J. H., Liquid-crystal phase equilibria of Lennard-Joneschains, accepted in Mol. Phys.

5. van Westen, T.; Oyarzún, B.; Vlugt, T. J. H.; Gross, J., An analytical equation of state for de-scribing isotropic-nematic phase equilibria of Lennard-Jones chain fluids with variable degreeof molecular flexibility, J. Chem. Phys. 142, 244903 (2015)

4. Oyarzún, B.; van Westen, T.; Vlugt, T. J. H., Isotropic-Nematic Phase Equilibria of Hard-Sphere Chain Fluids - Pure Components and Binary Mixtures, J. Chem. Phys. 142, 064903(2015)

3. van Westen, T.; Oyarzún, B.; Vlugt, T. J. H.; Gross, J., An equation of state for the isotropicphase of linear, partially flexible and fully flexible tangent hard-sphere chain fluids, Mol. Phys.112, 919–928 (2014)

2. Oyarzún, B.; van Westen, T.; Vlugt, T. J. H., The phase behavior of linear and partially flexi-ble hard-sphere chain fluids and the solubility of hard spheres in hard-sphere chain fluids, J.Chem. Phys. 138, 204905 (2013)

1. van Westen, T.; Oyarzún, B.; Vlugt, T. J. H.; Gross, J., The isotropic-nematic phase equili-brium of tangent hard-sphere chain fluids - Pure components, J. Chem. Phys. 139, 034505(2013)

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Acknowledgement

The decision of starting a PhD is usually a choice driven by curiosity in science connectedwith a personal commitment on understanding reality in its organic form. This romanticvision is confronted with the “reality” of science that reduces scientific progress to relevanceand quantity of the scientific output. Luckily, doing a PhD was for me the result of a healthybalance between both, a romantic vision that transforms curiosity into understanding and alarge amount of work that converts findings into a consistent scientific contribution.

The book that you are having now on your hands is the result of multiple hours of pro-gramming (mostly finding the bug!) and simulation runs that made possible the numericalresults presented in the main part of the text. Nevertheless, the meaning of these resultswould not be complete without the input, discussion and contribution of the exceptionalindividuals that enriched the vision and work presented here.

The relevance of this scientific work was mainly possible due to the input and expe-rience of my supervisor Thijs Vlugt. ThijsV, your absolute commitment to science was adecisive factor on making progress. I felt very lucky of having the advice (and here I haveto emphasize, almost immediate advice!) of somebody that knew precisely what to do atbifurcations and dead-end roads.

Coherence in this book is, in a large part, a consequence of the collaboration with Thijsvan Westen. ThijsvW, many of the conclusions that appear in this thesis are a result of ourdiscussions (after polishing some stubbornness from both sides). Comparing simulationresults with theoretical predictions did not only “match the lines with the points” but alsogave us a deeper understanding of the physics of fluids.

Programming and performing simulations have many tricks and ways or, better to say,elegant solutions to specific requirements. I would like to thank Sayee Prasaad Balaji andSondre Kvalvåg Schnell for their help in making the coding and simulation work muchsimpler.

I would also like to thank all the people involved in the liquid crystals project, speciallyMariëtte de Groen and Theo de Loos for all the discussions about experiments and thebehavior or real liquid crystal fluids.

The vision and motivation that made this book possible is strongly related to all thepeople with which I have shared this PhD path. I find very difficult now to write one namebefore another, because everybody had a special contribution in shaping this work and Ihope to find the words for expressing my gratitude to all of you (including the ones that arenot named here):

Irina, you are the person that has been closer to me during this process. You know thedifficulties and rewards that came across, and I had the fortune to be accompanied by youduring this time. Your presence gave me the energy to bring this book to an end, and itis also the inspiration for the dreams and facts that have came, and will come, along. Iappreciate all your patience and efforts while this thesis was taking form, while, somehow,everything else was taking form. Sunt foarte bucuros ca suntem aici unul pentru celalalt!. Iam very thankful to you for the design of the cover (I feel that you read my mind!), makinga beautiful and meaningful image for presenting this work.

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102 Summary

ThijsvW, apart from all the science related work said above, I have to say that it wasgreat to have you as a colleague. From the first talk that we had that influenced me intaking the decision of joining this project, until the last discussions about “... and nowwhat?”, I found always a good person to talk and to be listen. Trips were amazing, beer wasrefreshing, and always with fine music.

Sara, Jorge, Marta, I think that you are the ones that I know since my very first daysin Holland. Marta, many thanks for all the favors including letting me to be inscribed atyour place when I was starting to deal with the dutch bureaucracy. I am very happy that Icould showed you the place where I come from (including an earthquake magnitude 8.2 andmany 7+, which in any case are definitely an experience to remember!). Sara and Jorge,many thanks for all the talks, dinners, bike trips and specially for had taking part of yourwonderful wedding. Sara, Jorge and Marta, many thanks for your friendship.

Alex, you have been always been there, man. It is difficult to find somebody that it iswilling to meet or have a beer just after a call. Thanks for all the talking and joking around.

Shouruk (Su), I am very happy that we were flat-mates keeping always good spirits athome. Many thanks for watering the plants :p and all the other things that I could have notdone just by myself.

Dragos, without you I could have not met the girl that I love!. Many thanks for all thetime spent together, and for being always available when your help was needed.

Stevia, I want to thank you for your humor and advices. I will always admire yourcapacity of knowing what-to-do as cheaper-as-possible.

Sebastián, many thanks for all the talks with some “chelitas” at the side. I was happyof meeting somebody with who I could share the feeling of coming from the middle of theworld.

I would also like to thank everybody in the office, the old and new generations: Sayee(the boss!), Samir (life is complicated), Albert (lunch!), Lawien, Carsten, Mahinder, Emili-ano, Alondra/Timo, Sergio, Stephanie, Tim, Karsten, Julia, Dion and Daniel. Marloes, youand your social skills are definitely an important contribution to the department.

I would also like to thank Brian Tighe with who I had the privilege to work as teachingassistant during the Thermodynamics for Process & Energy course.

And also, I would like to thank all of you that have been far but at the same timevery close, Katharina, Diego, José, Marta, María, Sebastián/Simona, Steffi, Andrea, Nico,Dieguito, Sandra and Juanito.

Y finalmente, Papá/mamá/ñaña, esta tesis está dedicada a ustedes. El que haya llegado aseguir y a terminar un programa de doctorado es todo consecuencia del apoyo incondicionalde su parte. Ustedes siempre creyeron en mi y que podría cumplir cualquier proyecto con elque soñara. A ustedes le debo la visión y las ganas de explorar que me han llevado a sitiostan lejanos. Y aunque la distancia siempre nos ha dolido, creo que hemos podido encontrarla manera de estar cerca en todos estos años.